
MAURER 



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AM ERIC A^ 



C H } C A C J O 



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Class 
Book 



-A_40i 



^Al£___ 



Copyright ]^^ 



COPYRIGHT DEPOSIT, 



Strensfth of Materials 



A Practical Manual of 
SCIENTIFIC METHODS OF LOCATING AND DETERMINING STRESSES AND 
CALCULATING THE REQUIRED STRENGTH AND DIMEN- 
SIONS OF BUILDING MATERIALS 



By EDWARD R. MAURER, B.C.E. 

Professor of Mechanics, Tniversity of Wisconsin 



ILLUSTRATED 



CHICAGO 
AMERICAN SCHOOL Ol' lOKRllSroN I>1' NCE 






VuBHARY of congress] 

I NOV 2 s^^ 

1 Copynfht Entw 

GLASS A XXc„ NO. 
COPY S._ 



Copyright 1907 by 
American Schooi, of Cokj^espondence 



Kiitered at Stationers' Hall, London 
All Rig-hts Reserved 



• • • 

• • • 



Foreword 




N recent years, such marvelous advances liave beeii 
made in the engineering and scientific fields, and 
so rapid has been the evolution of mechanical and 
constructive processes and methods, that a distinct 
need has been created for a series of i^rncficnl 
vjorkliKj (juldcH^ of convenient size and low cost, embodying the 
accumulated results of experience and the most approved modern 
practice along a great variety of lines. To till this acknowledged 
need, is the special purpose of the series of handl)ooks to which 
this volume belongs. 

C 111 the preparation of this series, it has been the aim of the pub- 
lishers to lay special stress on the practical side of each subject, 
as distinifuished from mere theoretical or academic discussion. 
Each N'oluim^ is written by a well -known ('Xj)ert of acknowledged 
authority in liis special line, and is based on a most careful stndv 
of practical needs and up-to-date methods as dexclojx'd nnder the 
conditions of actual ])ractice in the field, tin* shop, llu^ mill, the 
power hous(^, the drafting room, the engine room, etc. 

C These Noluiiu'S are es|>ecially ada])ted for purposes of self- 
instruction and honu^ study. The utmost can* lias been us^mI to 
brinn" the t i'c;il iiiciil ol each subject within (he ranm- of llicroin- 



mon understanding, so that the work will appeal not only to the 
technically trained expert, but also to the beginner and the self- 
taught practical man who wishes to keep abreast of modern 
progress. The language is simple and clear; heavy technical terms 
and the formulae of the higher mathematics have been avoided, 
yet without sacrificing any of the requirements of practical 
instruction; the arrangement of matter is such as to carry the 
reader along by easy steps to complete mastery of each subject; 
frequent examples for practice are given, to enable the reader to 
test his knowledge and make it a permanent possession; and the 
illustrations are selected with the greatest care to supplement and 
make clear the references in the text. 

C The method adopted in the preparation of these volumes is that 
which the American School of Correspondence has developed and 
employed so successfully for many years. It is not an experiment, 
but has stood the severest of all tests — that of practical use — which 
has demonstrated it to be the best method yet devised for the 
education of the busy working man. 

C Foi' purposes of ready reference and timely information when 
needed, it is believed that this series of handbooks will be found to 
meet every requirement. 




Table of Contents 



Simple Stress . . . ' Page 

Kinds of Stress (Tension, Compression, Shear) — Unit-Stress — Defor- 
mation — Elasticity — Hooke's Law and Elastic Limit — Ultimate 
Strength — Stress-Deformation Diagram — Working Stress and 
Strength — Factor of Safety — Strength of Timber, Wrought Iron, Steel, 
Cast Iron, Brick, Stone, under Tension, Compression, Shear. 



Reactions of Supports Page 18 

Moment of a Force — Principle of Moments — Kinds of Bc'ams (Simple, 
Cantilever, Restrained, Continuous) — Determination of Reactions on 
Beams — External Shear and Bending Moment — Rule of Signs — Sliear 
Diagrams — Maximum Shear — Moment Diagrams — Maximum Bending 
Moment — Center of Gravity of Built-Up Sections — Moment of Inertia — 
Reduction Formula — Moment of Inertia of a Rectangle; of Built-Up 
Sections — Section Moduli — Radii of Gyration. 



Strength of Beams Page 54 

Kinds of Loads (Transverse, Longitudinal. Inclined Forces) — Neutral 
Surface — Neutral Line — Neutral Axis — Stress at Cross-Section — Fibre 
Stress — Resisting Moment — Beam FormuljT? and Applications — Prop- 
erties of Standard I-Beams — Laws of Strength of Beams (Safe 
Strength, Safe Load) — Modulus of Rupture — Resisting Shear — Hori- 
zontal Shear — Design of Timber Beams — Flexure and Tension — Flex- 
ure and Compression — Combined Flexural and Direct Stress. 



Strength of Columns . . . Page 87 

Columns, Posts, or Struts and Short Blocks — End Conditions (Hinged 
or Pin, Flat or Square, Fixed) — Cross-Sections of Columns — Radius 
of Gyration — Column Loads (Central, Eccentric) — Rankine's Column 
l<""ormula — Graphical Representation of Column Formulix' — Combination 
Column Formula* — Straight-Line and Euler Formula^ — Data for ^lild- 
Steel Columns — Parabola-Euler l<'onnuliv — Broken Straiglit-Llnc Fov- 
uiuhv — Design of Columns. 



Strength of Shafts, Riveted Joints, etc Page 107 

Twisting Moment — Torsional Sd'ess- — ii(>sisting Mttnicut — I'^onnula for 
Strcjigtli of Siiafl; for Powci- wiiicli Shall can 'i'ransmil — StilTn<>ss nf 
Rods, lieains, and Shafts — Coenicieiits of lOlastiri t .\' — Trmi.>era t ure 
Stresses- -lOloiigalidu ni- IJiwar I'iXpa iisiun 1 »<"lli(t imi of I'^ams - 
Twist of Shafts — Angle of Twisl -Non-Klastic Di-formaiitin -lii\oting 
• -Pitcli — Sliearing Strength of liivets — l^eariiig SlicngMi of Platt^s- 
l''rictional Strength of Joints — Tensile ami ( 'Dinpressive Strength of 
Illvotod Plates — Computing Strength nf a .Foint I'ltiniate and Work- 
ing EfHclency of .loints. 



Index . . . . Page 125 




LA SALLE STATION, L. S. & M. S. AND C. R. L & P. RAILROADS, CHICAGO, ILL. 

Frost & Granger, Architects; E. C. & R. M. Shankland, Engineers. 

Steel Frame Showing Method of Attaching Angle-Iron Shelf to Spandrel Beam. This Occurs at 

Each Floor Line, and Supports the Brick Walls. Note the Cast-Iron Supports in Window 

MuUions to Stiffen the Window Opening. Sometimes these ai'e Made of SteeL 

They are Fastened to the Floor Beams at Top and Bottom. 



STRENGTH OF MATERIALS. 



PART T. 



A 



^ 



I — I 



3 



A 



T 



SIMPLE STRESS. 

I. Stress. When forces are applied to a body they tend in a 
greater or less degree to break it.^ Preventing or tending to pre- 
vent the rupture, there arise, generally, forces between every two 
adjacent parts of the body. Thus, when a 
load is suspended by means of an iron rod, 
tl*e rod is subjected to a downward pull at 
its lower end and to an upward pull at its 
upper end, and these two forces tend to pull 
it apart. At any cross -section of the rod 
the iron on either side ''holds fast" to that on 
the other, and these forces which the parts 
of the rod exert upon each other prevent 
the tearing of the rod. For example, in Fig. 
1, let (C represent the rod and its suspended 
load, 1,000 pounds; then the pull on the 
lower end equals 1,000 ])ounds. If we neg- 
lect the weight of the rod, the ])ull on the 
upper end is also 1,000 pounds, as shown in 
Fig. 1 (//j; and the upj)er j)art A exerts 
on th(^ lower juirt ]> an upward j)ull Q (Mpial 
to 1,000 j)onn(ls, while the lower part exerts 
on the U[)j)CH' a force P also e(jual to .1,(M)() jjounds. Tliesi^ two 
forces, P and Q, prevent ruptui'o of tlui rod at {\\o ''scctioiT' ('; at 
any other section thei'o are two forces like P and (>), j»reventing 
ruptui-e at that section. 

P>Y .s7/v.s',s' (/f, ((' .serf ion of a body is meant tlie foi'ce which tli(> 
j>Hrt of tlie body on either si(h( of th(^ section c xiTts on th(> otht-i*. 
Thus, the stress at, tlu^ siH'tion (! (I'ig. 1) is P (or (,)), and it eipuils 
1,000 pounds. 

3. Sti"('ss(*s arc usually expressed (in Anu^rica) in n()unds, 
somelijnes in tons. Thus the sti-ess P \\\ lh«^ nrrccdiun- artich^ is 



^ 



V 



&«. 



Fi^^ 1. 



STRENGTH OF MATERIALS 



1,000 pounds, or -J ton. Notice that tliis value lias nothing to do 
with the size of the cross-section on which the stress acts. 

3. Kinds of Stress, (a) "When the forces acting on a body 
(as a rope or rod) are such that they tend to tear it, the stress at 
any cross-section is called a tension or a tensile stress. The 
stresses P and Q, of Fig. 1, are tensile stresses. Stretched ropes, 
loaded "tie rods" of roofs and bridges, etc., are under tensile stress, 
(b.) When the forces acting on a body (as a short post, brick, 

etc.) are such that they tend to 
crush it, the stress at any sec- 
tion at ricrht ancrles to the di- 
rection of the crushincr forces is 
called a pressure or a compres- 
sive stress. Fig. 2 (a) repre- 
sents a loaded post, and Fig. 2 
(^) the upper and lower parts. 
The upper part presses down on 
B, and the lower part presses up 
on A, as shown. P or Q is 
the compressive stress in the 
post at section C. Loaded posts, 
or struts, piers, etc., are under 
compressive stress. 

(c.) When the forces acting 
on a body (as a rivet in a bridge 
joint) are such that they tend to cut or " shear " it across, the stress 
at a section along which there is a tendency to cut is called a sheccr 
or a shearing stress. This kijid of stress takes its name from the 
act of cutting Avith a pair of shears. In a material w^hich is being 
cut in this way, the stresses that are being " overcome " are shear- 
ing stresses. Fig. 3 (r/) represents a riveted joint, and Fig. 3 (/>») 
two parts of the rivet. The forces applied to the joint are such 
that A tends to slide to the left, and B to the right; then B exerts 
on A a force P toward the rip-ht, and A on B a force Q toward the 
left as shown. P or Q is the shearing stress in the rivet. 

Tensions, Compressions and Shears are called sinijjle stresses. 
"Forces may act upon a body so as to produce a combination of simple 
stresses on some section; such a combination is called a coTnplex 




Fig. 2. 



STRENGTH OF MATEEIALS 



stress. The stresses in beams are usually complex. There are other 
terms used to describe stress; they will be defined farther on. 

4. Unit=Stress. It is often necessary to specify not merely 
the amount of the entire stress which acts on an area, but also the 
amount which acts on each unit of area (square inch for example). 
By unit-stress is meant stress per unit area. 

To find the value of a unit-stress: Divide the vjJtole stress hy 
the whole area of the section 0)i which it acts, or over which it is 
distributed. Thus, let 

P denote the value of the whole stress, 

A the area on which it acts, and 

S the value of the unit-stress; then 

P 

S=-^, alsoP-=AS. (i) 

Strictly these formulas apply only when the stress P is uniform, 




Fig. 3. 

that is, when it is uniformly distributed over the area, each square 
inch for example sustaining the same amount of stress. When 
the stress is not uniform, that is, when the stresses on different 
square inches are not e(]^ual, then P-^-A equals the avcnKje vat ho 
of the unit-stress. 

5. Unit-stresses are usually expressed (iu Americ;;) in 
pounds per square inch, sometimes in tons per square incli. ]f 
P and A in equation 1 are expressed in pounds and S([u:ir(' 
inches respectively, then B will be in pounds per S(|uare inch; .-md 
if P and A are expressed in tons and square inches, S will bo in 
tons per square inch. 

Exrfi/ijdc.^. 1. Suppose that the rod sustaining- the load in 
Fig. 1 is 2 S(piare inches in cross-section, and that the load weighs 
1.000 pounds. What is the value of tlu' unit-stress ? 



6 STEEXGTH OF :\IATEPJALS 

Here P = 1.000 pounds, A= 2 square inches; hence. 

1.000 

S = — ^ — = 500 pounds per square inch. 

2. Suppose that the rod is one-half square inch in cross-sec- 
tion. AVhat is the value of the unit-stress ? 

1 

A = -;y-square inch, and, as before, P= 1,000 pounds; hence 

S = 1,000-f--;^ = 2,000 pounds pjer square inch. 

Xotice that one must always divide the whole stress bv the area to get 
the unit-stress, whether the area is greater or less than one. 

6. Deformation. ^Mienever forces are applied to a body it 
changes in size, and usually in shape also. This change of size 

*and' shape is called deformation. Deformations are iisually meas- 
ured in inches; thus, if a rod is stretched 2 inches, the '-elono-a- 
tioii'*= 2 inches. ♦ 

7. Unit-Deformation. It is sometimes necessary to specify 
not merely the value of a total deformation but its amount per 
unit leno'th of the deformed body. Deformation per unit length 
of the deformed body is called unit-deformation. 

To find the value of a unit-deformation: Divide the v:JioIe 
deformation ly tie length over which it is distriJjiited. Thus, if 
D denotes the value of a deformation, 
I the lenerth, 
s the unit-deformation, then 

6=^-T-, also D = /.v. (2) 

Both D and I should always be expressed in the same unit. 

Ex<Jiii2}le, Suppose that a 4-foot rod is elongated \ inch. 
What is the value of the unit-deformation? 

Here D=i inch, and /=4: feet=lS inches; 
hence .5-::^4^^-18=:-Jg inch per inch. 
That is, each inch ^s elonD-ated.-nV inch. 

I nit-elonoations are sometimes expressed in ])er cent. To 
express an elongation in per cent: Divide the elongcitioih in inches 
Ijij the original length in inches, and raultijply hy 100. 

8. Elasticity. JMost solid bodies when deformed will regain 
more or less completely their natural size and shape when the de- 



STRENGTH OF MATERIALS 



forming forces cease to act. This property of regaining size and 
shape is called elasticity. 

We may classify bodies into kinds depending on the degree 
of elasticity wliicli they have, thus: 

1. Perfectly eJai^tJc bodies; tliese will regain their orig- 
inal form and size no matter how large the ap]:)lied forces are if 
less than breaking values. Strictly there are no such materials, 
but rubber, practically, is perfectly elastic. 

2. Im/perfectly elastic bodies; these will fully regain their 
original form and size if tlie applied forces are not too large, and 
practically even if the loads are large but less than the breaking 
value. Most of the constructive materials belont; to this class. 

3. Inelastic OY j:)] a stic\i(j(l\e^\ ihk^i^Q will not regain in the 
least their original form when the applied forces cease to act. ^"lay 
and putty are good examples of this class. 

9. Hooke's Law, and Elastic Limit. If a gradually increas- 
ing force is applied to a ])erfectly elastic inatei'ial, the deformation 
increases proportionally to the force; that is, if P and V denote 
two values of the force (or stress), and 1) and D' the values of the 
deformation produced l)y the force, 

thenP:F::D:D'. 

This relation is also true for imperfectly elastic materials, 
provided that the loads P and P' do not exceed a certain limit dej)end- 
infr on the material. Beyond this limit, the deformation increases 
much faster than the load; that is, if within the limit an addition 
of l,000])ounds to the load jiroduces a stretch of O.Ol inch, lH'\()iid 
the limit an e(jual addition produces a stretch larger and usually 
much lar O'er than 0.01 inch. 

Beyond this limit of ])rop()rtionality a jtai't of the deforiiintion 
is permanent; that is, if the load is j-eni()\-ed the boch onl y partially 
recovers its foiMn and si/(\ I'lu* pernianent part of a dcfonnat ion 
is called set. 

The fact that I'oi" most materials tlio deformation is propor- 
tional to the load within certain liiiiils, is known as llook','"s Law. 
The unit-stress ^yithin whicli Hooke's law holds, or above which 
tlie deformation is not pi'oportioiial to the load oi' sti'ess, is called 
4 (((stic limit. 



8 STKEXGTH OF MATEEIALS 

10. Ultimate Strength. By ultimate tensile, compressive, 
or shearing strength of a material is meant the greatest tensile, 
compressive, or shearing unit-stress which it can withstand. 

As before mentioned, when a material is subjected to an in- 
creasincr load the deformation increases faster than the load beyond 
the elastic limit, and much faster near the stage of rupture. Xot 
only do tension bars and compression blocks elongate and shorten 
respectively, but their cross-sectional areas change also; tension 
bars thin down and compression blocks ^-'swell out" more or less. 
The value of the ultimate strength for any material is ascertained 
by subjecting a specimen to a gradually increasing tensile, com- 
pressive, or shearing stress, as the case may be, until rupture oc- 
curs, and measuring the greatest load. The Ireahing load divided 
hy the area of the original cross-seetion .sustaining the stress, is the 
vahie of the ^ultimate strength. 

E^.rarnple. Suppose that in a tension test of a wrought- iron 
rod \ inch in diameter the greatest load was 12,540 pounds. "What 
is the value of the ultimate strencrth of that oTade of wrought iron? 

o o o 

The orio'inal area of the cross-section of the rod was 

0.7854 (diametery=0.7854x J=0.1964 square inches; hence 

che ultimate strength equals 

12,540-^-0. 1964=63, 850 pounds per square inch. 

11. Stress=Deformation Diagram. A ''test" to determine 
the elastic limit, ultimate strength, and other information in re- 
gard to a material is conducted by applying a gradually increasing 
load until the specimen is broken, and noting the deforination coi-- 
responding to many values of the load. The first and second col- 
umns of the following table are a record of a tension test on a steel 
rod one inch in diameter. The numbers in the first column are 
the values of the pull, or the loads, at which the elongation of 
the specimen was measured. The elongations are given in the sec- 
ond column. The numbers in the third and fourth columns are 
the values of the unit-stress and unit-elongation corresponding to 
the values of the load opposite to them. The numbers in the 
third column were obtained from those in the first by dividing 
the latter by the area of the cross-section of the rod. 0,7854 
square inches. Thus, 

3.930^0.7854=5.000 
7,850-^0.7854=10,000, ete. 



STEEXGTll OF MATEEIALS 



9 



1 


Total Pull 


Deformation 


Unit-Stress in 
pounds per 


Unit- 
Deformation, 


in pounds, P 


in inches, D 


square inch, S 


s 


3930 


0.00136 


5000 


0.00017 


7850 


.(J0280 


looa) 


.(X)035 


11780 


.00404 


15000 


.00050 


15710 


.00538 


2oajo 


.00067 


19635 


.00672 


2;'500O 


.00084 


23560 


.00805 


30oaj 


.00101 


27490 


.00942 


35000 


.00118 


31415 


.01080 


40000 


.00135 


35345 


.01221 


45000 


.00153 


39270 


.0144 


50000 


.00180 


43200 


.0800 


55000 


.0100 


47125 


.1622 


60000 


.0202 


51050 


.201 


65000 


.0251 


54980 


.281 


70000 


.0351 


58910 


.384 


75000 


.048 


62832 


.560 


800a) 


.070 


65200 


1.600 


83000 


.2(X) 



The numbers in tlie fourth column were obtained by dividing 
those in the second by the length of the specimen (or rather the 
length of that ])art whose elongation was measured), 8 inches. 
Thus, 

0.001 30 --8 = 0.00017, 
.00280-^8 = .00035, etc. 

Lookino; at the first two columns it will be seen that the eloncra- 
tions are practically proportional to the loads up to the ninth load, 
the increase of stretch for each increase in load beintr about 0.0013.") 
incli; but beyond the ninth load the increases of stretch are much 
greater. Hence the elastic limit was reached at al)out the ninth 
load, and its value is about 45,000 pounds per s(puire inch. The 
greatest load was 05,200 pounds, and the corresponding unit-stress, 
83,000 pounds per square inch, is the ultimate strength. 

IRearly all the information revealed by such a test can be 
well represented in a diagram called a HfrrHx-drforinafioih dnajiunn. 
It is made as follows: Lay off the values of the unit-deformation 
(fourth column) along a horizontal line, according to some con- 
venient scale, from some fixed ])oint in the line. At the points on 
the horizontal lino representinix the vario\ia unit-elontrations, lav 
oft* perpendicular distances ecpial to the corres])oiuling unit-stresses. 
Then connect by a smooth cui've tlio Uj)|)er ends of all those dis- 
tances, l:ist distances laid oil. Thus^ for instance, thi» hitrjiest unit- 



10 STEEXGTII OF MATERIALS 

elongation (0.20) laid off from >> (Fig. 4j fixes tlie point a^ and a 
perpendicular distance to represent the highest unit-stress (83,000) 
fixes the point Ij. All the points so laid off give the curve och. The 
part or^ within the elastic limit, is straight and nearly vertical 
while the remainder is curved and more or less horizontal, especially 
toward the point of rupture h. Fig. 5 is a tvpical stress -defor- 
mation diagram for timber, cast iron, wrought iron, soft and hard 
steel, in tension and compression. 

12. Working Stress and Strength, and Factor of Safety. 
The greatest unit-stress in any 2)art of a structure when it is sus- 

tainincf its loads is called the 

o 

worJi'tng stress of that part. If 
it is under tension, compression 
and shearing stresses, then the 
corresponding highest unit- 
stresses in it are called its work- 
incr stress in tension, in com- 
pression, and in shear respect- 
ively; that is, we speak of as 
^. , many workincr stresses as it has 

Fig. i, . " 

kinds of stress. 
By vjoj'Jihir/ streiKjili of a material to be used for a certain 
purpose is meant the highest unit-stress to which the material 
ought to be subjected when so used. Each material has a working 
strength for tension, for compression, and for shear, and they are in 
o-eneral different. 

^^ factor of safety i^ meant the ratio of the ultimate strength 
of a material to its workinof stress or streng^th. Thus, if 
S^ denotes ultimate strength, 
S,„ denotes workincr stress or streno-th, and 
y denotes factor of safety, then 

f=^;^\^o&^ = ^.. (3) 

When a structure which lias to stand certain loads is about 
to be designed, it is necessary to select working strengths or fac- 
tors of safety for the materials to be used. Often the selection is 
a inatter of great importance, and can be wisely performed only 
by an experienced engineer, for this is a matter where hard -and- 




STEENGTIL OF MATEPJALS 



11 



fast rules should not govern but rather the judgment of the expert. 
But there are certain principles to be used as guides in making a 
selectioUj chief among which are: 

1. The working strength should be considerably below the 
elastic limit. (Then the deformations will be small and not per- 
manent.) 




Fi^^ 5. (After Johnson.) 

2. The working strength should be smaller for parts of a 
structure sustainino; varyino- loads tlian for those whose loads are 
steady. (Actual experiments luive disch)se(l tin* fact tliat the 
strength of a Sj)ecimen (lepends on the kind of h)nd put uj)()n it, 
and tliat in a general way- it is less the less steady the load is.^ 

H. Tlie woi'kino' strenotli must l)e taken low I'oi- non uniform 
material, whei-e jioor workmansliip nuiy l)e expected, uIumi the 



12 



STEENGTH OF MATERIALS 



loads are uncertain, etc. Principles 1 and 2 have been reduced 
to figures or formulas for many particular cases, but the third must 
remain a subject for display of judgment, and even good guessing 
in many cases. 

The following is a table of factors of safety* which will be 
used in the problems: 

Factors of Safety. 



Materials. 


For steady 

stress. 
(Buildings.) 


For varying 

stress. 

(Bridges.) 


For shocks. . 
(Machines.) 


Timber 

Brick and stone 
Cast iron 
Wrought iron 
Steel 


8 
15 
6 
4 
5 


10 

25 
15 

6 

7 


15 

30 
20 
10 
15 



They must be regarded as coverage values and are not to be 
adopted in every case in practice. 

Exmnjples. 1. A wrought- iron rod 1 inch in diameter sus- 
tains a load of 30,000 pounds. What is its working stress? If 
its ultimate strength is 50,-000 pounds per square inch, what is 
its factor of safety % 

The area of the cross-section of the rod equals 0.7854 X (diam- 
eter)''-'=0.7854 X l'=0.7854 square inches. Since the whole stress 
on the cross-section is 30,000 pounds, equation 1 gives for the 
unit working stress 

.. 30,000 oo.nr. ^ • -u 

S = 7w%Fl ^^ 38,197 pounds per square mch. 

Equation 3 gives for factor of safety 

50,000 



f- 



1.3 



38,197 

2. How large a steel bar or rod is needed to sustain a steady 
pull of 100,000 pounds if the ultimate strength of the material is 
65,000 pounds ? 

The load being steady, we use a factor of safety of 5 (see table 
above); hence the working strength to be used (see equation 3) is 



S 



65,000 



13,000 pounds per square inch. 



The proper area of the cross -section of the rod can now be com- 
puted from equation 1 thus: 
•^Taken from Merriman's "Mechanics of Materials. ' 



STEENGTII OF MATERIALS 13 

P 100,000 „^^^ 
A ^^== 7^ = .... .,.,,, = 7.692 square inches. 

A bar 2x4 inches in cross-section would be a little stronger 

than necessary. To find the diameter (<:Z) of a round rod of sulii- 

cient strength, we write 0.7854 ^'Z^ = 7.G92, and solve the enuation 

iord; thus: 

7 692 
(V=.^ ' ^^ ■ = 9.794, or <:/== 8.129 inches. 

0.7854 ' 

3. How large a steady load can a short timber post safely sus- 
tain if it is 10x10 inches in cross-section and its ultimate com- 
pressive strength is 10,000 pounds per square inch ? 

According to the table (page 12) the proper factor of safety is 
8, and hence the working strength according to equation 3 is 

b = — r. — =^ l,2oU pounds per square men. 

Th^ area of the cross-section is 100 square inches; hence the safe 
load (see equation 1) is 

r = 100 X 1,250 -- 125,000 pounds. 

4. When a hole is punched through a ])late the shearing 
strength of the material has to be overcome. If the ultimate shear- 
ing strength is 50,000 pounds ])er square inch, the thickness of the 
])late i\ inch, and the diameter of the hole J inch, what is the value 
of the force to be overcome ? 

The area shorn is that of the cylindrical surface of tlie hole 
or the metal punched out; that is 

3.1416 X diameter X thickness = 3.1416 X ij X -1 =— 1.178 stp in. 
Hence, by equation 1, the total shearing strength or resistance 
to ]>unching is 

P -- 1.178 X 50,000 -= 58,900 pounds. 

STRENGTH OF MATERIALS UNDER SIMPLE 5TRE5S. 

i3. Mate ials in Tension. Pi'actically (lie onlv mntiM-Jals 
used extensively under tension are timber, wrought iron and steel, 
and to soim^ extent cast iron. 

14. Timber. A successful tension lest of \\()od is diliienll, 
as the sjHH'imen usually crushes at the entls \\lieii held in the tv'st- 
ing machine, splits, or fails otherwise than as desired. Hence the 



14 STEEIS^GTII OF MATEEIALS 

tensile strengths of woods are not well known, but the following 
may be taken as approximate average values of the ultimate 
strengths of the woods named, v/hen ^'drv out of doors." 

Hemlock, 7,030 pounds per square inch. 

White pine, 8,000 

Yellow pine, long leaf, 12,000 " " 

" " , short leaf, 10,000 " « 

Douglas spruce, 10,000 " " 

White oak, 12,000 " « 

Ked oak, 9,000 " 

15. Wrought Iron. The process of the manufacture of 
wroug-ht iron gives it a ''grain," and its tensile streno-ths alono- and 
across the grain are unequal, the latter being about three-fourths 
of the former. The ultimate tensile strength of wrought iron 
along the grain varies from 45,000 to 55,000 pounds per square 
inch. Strength along; the grain is meant when not otherwise 
stated. 

The strength depends on the size of the piece, it being greater 
for small than for large rods or bars, and also for thin than for 
thick plates. The elastic limit varies from 25,000 to 40,000 
pounds per square inch, depending on the size of the bar or plate 
even more than the ultimate strength. TTrought iron is very 
ductile, a specimen tested in tension to destruction elongatincT from 
5 to 25 per cent of its length. 

16. Steel. Steel has more or less of a grain but is 2:)ractically 
T)f the same strength in all directions. To suit different purposes, 
steel is made of various grades, chief among; which may be men- 
tioned rivet steel, sheet steel (for boilers), medium steel (for 
bridges and buildings) , rail steel, tool and spring steel. In general, 
these grades of steel are hard and strong in the order named, the 
ultimate tensile strength ranging from about 50,000 to 100,000 
pounds per square inch. 

There are several grades of structural steel, which may be 
described as follows:* 
1. Kivet steel: 

Ultimate tensile strength, 48,000 to 58,000 pounds per square inch. 
Elastic limit, not less than one-half the ultimate strength. 
Elongation, 26 per cent. 
Bends 180 degrees flat on itself without fracture. 



*Taken from " Manufacturer's Standard Specifications." 



STKE^^GTII OF MATERIALS 15 

2. Soft steel: 

Ultimate tensile strength, 52,000 to 62,00r) pounds per square inch. 
Elastic limit, not less than one-half the ultimate strength. 
Elongation, 25 i)cr cent. 
Bends ICO degrees flat on itself. 

3. Medium steel: 

Ultimate tensile strength, 60,0(X) to 70,000 ])ounds per S(iuarc inch. 
Elastic limit, not less than one-half the ultimate strength. 
Elongation, 22 per cent. 

Bends 180 degrees to a diameter equal to the thickness of the 
specimen without fracture. 

17. Cast Iron. As in tlie case of steel, there are many 
grades of cast iron. The grades are not the same for all localities 
or districts, but they are based on the appearance of the fractures, 
which vary from coarse dark grey to fine silvery white. 

The ultimate tensile strength does not vary uniformly with 
the grades but depends for the most part on the percentage of 
"combined carbon" present in the iron. This strength varies from 
15,000 to 35,000 pounds per square inch, 20,000 being a fair 
average. 

Cast iron has iio well-defined elastic limit (see curve for cast 
iron, Fig. 5). Its ultimate elongation is about one per cent. 

EXAMPLES FOR PRACTICE. 

1. A steel wire is one-eighth inch in diameter, and tlie ulti- 
mate tensile strength of the material is 150,000 pounds per s(puire 
inch. How large is its breaking load ? Ans. 1,810 pounds. 

2. A wroucrht-iron rod (ultimate tensile streno-th 50,000 
pounds j)er square inch) is 2 inches in diameter. How lai-ge a 
steady pull can it safely l)ear ? Ans. BU,2T0 ])ounds. 

18. Materials in Compression. T'nlike the tensile, the 
compressive strengtli of a specimen or structural part deju'iids on 
its dimension in tht^ dii'ection in ^\hi('ll tlie load is aj)[)Hed, for, 
in compression, a long bar oi* rod is weaker than a short one. A I 
present we refer only to llic! streiiglh of slioi-t pii'ces such as do 
r.ot Ix'IkI under the h)ad, the longer ones (ci)luinns) bi'ine; dis- 
cussed farthei" on. 

Did'ei'ent nialei'ials l»i'eak or fail undei* i-oni pression, in two 
V(uy dilTereiit ^\a^^s: 

I. 1 )u('t ile niatei-ials (structural steel, wrought inui, etc.), 



16 STEE]SrGTH OF MATEEIALS 



and woad compressed across the grain, do not fail by breaking into 
two distinct parts as in tension, but tbe former bulge out and 
flatten under great loads, while wood splits and mashes down. 
There is no particular point or instant of failure under increasino- 
loads, and such materials have no definite ultimate streno-th in 
compression. 

2. Brittle materials (brick, stone, hard steel, cast iron, etc.), 
and wood compressed along the grain, do not mash gradually, but 
fail suddenly and have a definite ultimate strength in compression. 
Although the surfaces of fracture are always much inclined to the 
direction in which tlie load is applied (about 45 degrees), the ulti- 
mate strength is computed by dividing the total breaking load by 
the cross -sectional area of the specimen. 

The principal materials used under compression in structural 
work are timber, wrought iron, steel, cast iron, brick and stone. 

19. Timber. As before noted, timber has no definite ulti- 
mate compressive strength across the grain. The U. S. Forestry 
Division has adopted certain amounts of compressive deformation 
as marking stages of failure. Three per cent compression is 
regarded as "a working limit allowable," and fifteen per cent as 
"an extreme limit, or as failure." The following (except the first) 
are values for compressive strength from the Forestry Division 
lleports, all in pounds per square inch: 

Ultimate strenjofth 3:^ Compression 

along the grain. across the grain 

Hemlock........ 6,000 

White pine 5,i00 700 

Long-leaf yellow pine 8,003 1,260 

Short-leaf yellow pine 6,500 1,050 

Douglas spruce 5,700 800 

White oak 8,500 2,200 

Red oak 7,200 2,300 

20. Wrought Iron, The elastic limit of wrought iron, as be- 
fore noted, depends very much upon the size of the bars or plate, it 
being greater for small bars and thin plates. Its value for com- 
pression is practically the same as for tension, 25,000 to 40,000 
pounds per square inch. 

21. Steel. The hard steels have the liighest compressive 
strength; there is a recorded value of nearly 400,000 pounds per 
square inch, but 150,000 is probably a fair average. 



STKENGTH OF MATERIALS 17 

The elastic limit in compression is practically the same as in 
tension, which is about 60 per cent of the ultimate tensile strength, 
or, for structural steel, about 25,000 to 42,000 pounds per square 

inch. 

22. Cast Iron. This is a very strong material in compres- 
sion, in which way, principally, it is used structurally. Its ulti- 
mate strength depends much on the proportion of "combined car- 
bon" and silicon present, and varies from 50,000 to 200,000 pounds 
per square inch, 90,000 being a fair average. As in tension, 
there is no well-defined elastic limit in compression (see curve for 
cast iron, Fig. 5). 

23. Brick. The ultimate strengths are as various as the 
kinds and makes of brick. For soft brick, the ultimate strencrth 
is as low as 500 pounds per square inch, and for pressed brick it 
varies from 4,000 to 20,000 pounds per square inch, 8,000 to 
10,000 being a fair average. The ultimate strength of good pav- 
ing; brick is still hio-her, its averao;e value being; from 12,000 to 
15,000 pounds per square inch. 

24. Stone. Sandstone, limestone and granite are the 
principal building stones. Their ultimate strengths in pounds 
per square inch are about as follows: 

Sandstone * 5,000 to 16,000, average 8,000. 
Limestone,* 8,000 " 16,000, " 10,000. 
Granite, 14,000 " 24,000, " 16,000. 
•^Compression at right angles to the "bed" of the stone. 

EXAMPLES FOR PRACTICE. 

1. A limestone 12x12 inches on its bed is used as a pier 
cap, and bears a load of 120,000 pounds. Wiat is its factor of 
safety? Ans. 12. 

2. How large a post (short) is needed to sustain a steady 
load of 100,000 pounds if the ultimate compressive strength of 
the wood is 10,000 pounds per square inch ? Ans. lOX 10 inches. 

25. Materials in Shear. The principal materials used uiuIit 
shearing stress are timber, wrought iron, steel and cust iron. 
Pai-lly on account of (ln^ dilliculty of (leti'i'inining slicaring 
strengtlis, these ;ire not well known. 

26. Timber. Tlie ultini:i((^ .^Iicuimik'* stri'inrths of llu< more 
important woods (iloiuj the (jniHi are about as foHows: 



18 STRE^^GTH OF MATERIALS 

Hemlock, 300 pounds per square inch. 

White pine, 400 " 

Long-leaf yellow i:)ine, 850 " 

Short-leaf " " 775 " 

Douglas spruce, 500 " 

White oak, 1,000 « 

Red oak, 1,100 « 

Wood rarely fails by shearing across the grain. Its ultimate 



Y^ 



P 

Fig. 6 a. Fig. 6 d. 

shearing strength in that direction is probabh" four or five times 
the values above given. 

27. Metals. The ultimate shearing; strength of wrought 
iron, steel, and cast iron is about 80 per cent of their respective 
ultimate tensile strengths. 

EXAMPLES FOR PRACTICE. 

1. How large a pressure P (Eig. 6 <r^) exerted on the shaded 
area can the timber stand before it will shear off on the surface 
ahcd, if ah = 6 inches and he = 10 inches, and the ultimate shear- 
ing strength of the timber is 400 pounds per square inch ? 

Ans. 21,000 pounds. 

2. "When a bolt is under tension, there is a tendency to tear 
the bolt and to ''strip", or shear off the head. The shorn area 
would be the surface of the cylindrical hole left in the head. 
Compute the tensile and shearing unit-stresses when P (Fig. />) 
equals 30,000 pounds, (7 = 2 inches, and t = 3 inches. 

( Tensile unit-stress, 9,550 pounds per square inch. 
' ( Shearing unit-stress, 1,595 pounds per square inch. 

REACTIONS OF SUPPORTS. 

28. Moment of a Force. By moment of a force with re- 
spect to a point is meant its tendency to produce rotation about 
that point. Evidently the tendency depends on the magnitude of 
the force and on the perpendicular distance of the line of action 
of the force from the point : the greater the force and the per- 
pendicular distance, the greater the tendency; hence the mojneut 




< 
Z 

o 

(J 
U 
(O 

UJ 

tn 
q: 
u 

> 

z 
a: 



] 



t^ 




^ iT 



/ I 



z 
o 

o 

u 



< 

z 

o 

o 

z 
o 



o 
o 

i-J 6 
fa ^ 

fa .*- 

o s 

en ^ 
W ^• 

gi 

o ^ 

K I 
O C 

O i 

w I: 

w •^ 

O "tr 

a ^ 
w ^ 

o ^ 

EC 5^ 

O ^ 

fa --v^ 

Z ^' 

5 ?- 

§.? 

O ft. 

^^ 'u 

U 
fa 
(A 



:i77jr 



STEENGTII OF MATERIALS 



19 



of a force with resjject to a jpoint eqtoals the product (f the force 
and the p}(ii^p<i'>^dicnlar distance from the force to the point. 

The point with respect to which the moment of one or more 
forces is taken is called an origin or center of moments^ and the 
perpendicular distance from an origin of moments to the line of 
action of a force is called the arm of the force with respect to 
that origin. Thus, if Fj and Fg (Fig. 7) are forces, their arms 
with respect to O' are a^ and a.^' respectively, and their moments 
are Y/f\ and F/^;'^. With respect to 0'' their arms are c^' and a" 
respectively, and their moments are F/^/' and F.// ,". 

If the force is expressed in pounds and its arm in feet, the 
moment is in foot-pounds; if the force is in pounds and the arm 
in inches, the moment is in inch-pounds. 

29. A sign is given to the moment of a force for conven- 
ience; the rule used herein is as follows: The moment of a 
force about a poiiLt is p>ositive or negative according as it tends 
to turn the hody ahout that point in the clochwise or counter- 
clochwise direction'^ . 
Thus the moment (Fig. 7) 

of F, about ()' is negative, about O" positive; 



a 



30c Principle of Moments. 

proper magnitude and line of ac- 
tion can balance any number of 
forces. That sincrle force is called 
the equilihrant of the forces, and 
the single force that would balance 
the equilihrant is called the result- 
ant oi the forces. Or, otherwise 
stated, the resultant of any num- 
ber of forces is a force which pro- *^i 
duces the same effect. It can be 
proved that — The algehraic sum 
(f the mrnucuts ff <fnf/ nuiidKr 
(f forces ipilji rcsjK'ct to <f J>i>/nf^ 
('{jiK/ls the iiH>iiit))t of fht'/r I't'- 
sulfunf (ihoiif that jxiiiif. 



, about ()" neorative. 



In general, a sino;le force of 




l!4'. i 



*Ry clockwise^ dircvtion is iiu>aiit that in whirli tht» liaiiiN of a i-lorU 
rotut(s and by counUn- clorkvviso. th("i opposite -tliriH'tiorK 



20 



STEENGTH OF MATEKIALS 



This is a useful principle and is called "principle of moments." 
31. All the forces acting upon a body which is at rest are 

said to be halanced or in equilihrium. ]N"o force is required to 
balance such forces and hence their equilibrant and resultant are 
zero. 

Since their resultant is zero, the algebraic sum of tlie mom- 



looollos. aooolbs. 



aooolbs. 



looolbs. 



4.' — 'H-a'-^^ 



T 



D 



12' 



?^a-¥ 



aooolbs. 



■^ 



l- 



i, 



aooolbs. 



Fiff. 8. 



ents of any nuinber of forces wliicli are halccnced or in equilib- 
riuTYb equals zero. 

This is known as the principle of moments for forces in 
equilibrium; for brevity we shall call it also "the principle of 
moments." 

The principle is easily verified in a simple case. Thus, let 
AB (Fig. 8) be a beam resting on supports at C and F. It is 
evident from the symmetry of the loading that each reaction 
equals one-half of the whole load, that is, \ of 6,000=3,000 
pounds. (We neglect the weight of the beam for simplicity.) 

With respect to C, for example, the moments of the forces 
are, takincr them in order from the left: 

—1,000 X 4 =— 4,000 foot-pounds 

* 3,000 X 0= " 

2,000 X 2= 4,000 

2,000X14= 28,000 

— 3,aX) X 16 = — 48,000 

1,000X20= 20,000 

The algebraic sum -of these moments is seen to equal zero. 

Again, with respect to B the moments are: 

— 1,000 X 24 = — 24,000 foot-pounds 
3,000X20= 60,000 

— 2,000 X 18 = —36,000 

— 2,000 X 6 = — 12,000 
3,000 X' 4= 12,000 
1,000 X 0= " 

The sum of these moments also equals zero. In fact, no matter 



STEEN^GTTI OF MATERIALS 21 

where the center of moments is taken, it will be found in this and 
any other balanced system of forces that the algebraic sum of their 
moments equals zero. The chief use that we shall make of this 
principle is in finding the supporting forces of loaded beams. 

32. Kinds of Beams. A cantilever }>eam is one resting on 
one support or fixed at one end, as in a wall, the other end being 
free. 

A simjyle heain is one resting on two supports. 

A restrained Ijeam is one fixed at both ends; a beam fixed at 
one end and resting on a support at the other is said to be re- 
strained at the fixed end and simply supported at the other. 

A continuous heam is one resting on more than two supports. 

2tZ' Determination of Reactions on Beams. The forces which 
the supports exert on a beam, that is, the ''supporting forces," are 
called reactions. We shall deal chiefly with simple beams. The 
reaction on a cantilever beam supported at one point evidently 
equals the total load on the beam. 

When the loads on a horizontal beam are all vertical (and 

1000 lbs. zooplbs. 3000 lbs. 



A-x 



E) CD 



Fi^. 9. 



this is the usual case), the supporting forces are also vertical and 
the sam of the reactions equals the sum of the loads. This ])rin- 
ciple is sometimes useful in determining reactions, but in the case 
of simple beams the principle of moments is sufticient. The gen- 
eral method of determinino; reactions is as follows: 

1. Write out two e(|uati()ns of moments for all tlu' forces 
(loads and reactions) acting on the beam with origins of moments 
at the su[)j)orts. 

2. Solve the e(juations for the reactions. 

)l As a clieck, try if tlie sum of the ri'actions eipials the 
sum of the loads. 

K,raiu]des. 1, i"ig. 1) i-epi-escnts a beam su pporti'd at its 
ends and sustainintjf three loads. AV'^o wish to find the reaotiona 
duo to these loads. 



90 



STRENGTH OJb MATERIALS 



Let tlie reactions be denoted by R^ and R^ as shown; then 
the moment equations are: 
For origin at A, 

1,000 X 1 + 2,000 X 6 -4- 3,000 X8~-R,,X 10 = 0. 
For origin at E, 



2ioolbs. 

2'- 



3600 llOS. 



leoolbs. 



B 



6'- 



.^ 



&'- 



-4'- 



Fig. 10. 

R, X 10—1,000 X 9—2,000 X 4—3,000 X 2 = 0. 
The first equation reduces to 

10 R, = 1,000+12,000 + 24,000 = 37,000; or 
Rg^ 3,700 pounds. 
The second equation reduces to 

10 R = 9,000 + 8,000 + 6,000 = 23,000; or 
Rj= 2,300 pounds. 
The sum of the loads is 6,000 pounds and the sum of the reactions 
is the same; hence the computation is correct. 

2. Fig. 10 represents a beam supported at B and D (that is, 
it has oyerhancrincr ends) and sustaininor three loads as shown. We 
wish to determine the reactions due to the loads. 

Let Rj and R^ denote the reactions as shown ; then the moment 
equations are: 
For origin at B, 

-2,100x2 + + 3,600x6— R_,Xl4i-l,600xl8 --0. 
For origin at D, 

- 2,100xl6 + R,X 14— 3,600x8 + + 1,600x4 = 0. 
The first equation reduces to 

14 R,= -4,200 + 21,600 + 28,800 = 46,200; or 
R2 = 3,300 pounds. 
The second equation reduces to 

14 R,= 33,600 + 28,800-6,400 = 56,000; or 
Rj= 4,000 pounds. 
The sum of the loads equals 7,300 pounds and the sum oi the 
reactions is the same; hence the computation checks. 

3. What are the total reactions in example 1 if the beam 
weighs 400 pounds ? 



STRENGTH OF MATERIALS 23 

(1.) Since we already know the reactions due to the loads 
(2,300 and 3,700 pounds at the left and right ends respectively 
(see illustration 1 above), we need only to compute the reactions 
due to the weight of the beam and add. Evidently the reactions 
due to the weight equal 200 pounds each; hence the 

left reaction =2,300 + 200=2,500 pounds, and the 
right " -=3,700 + 200-=3,l)00 " 

(2.) Or, we might compute the reactions due to the loads 
and weight of the beam together and directly. In figuring the 
moment due to the weight of the beam, we imagine the weight 
as concentrated at the middle of the beam; then its mom.ents with 
respect to the left and right supports are (400 X 5) and — (400 X 5) 
respectively. The moment equations for origins at A and E are 
like those of illustration 1 except that they contain one more 
term, the moment due to the weight; thus they are respectively: 

1,000 X 1 + 2,000 X 6 + 3,000 X 8— E^, X 10 + 400 X 5 -=0, 

Ri X 10—1,000 X 9—2,000 X 4—3,000 X 2—400 X 5-=0. 
The first one reduces to 

10 R,= 39,000, or R.. = 3,900 ])ounds; 

and the second to 

10 R = 25,000, or R,= 2,500 pounds. 

4. What are the total reactions in examjJe 2 if the beam 
weighs 42 pounds per foot ? 

As in example 3, we might compute the reactions due to the 
weight and then add them to the corresponding reactions due to 
the loads (already found in example 2), but we shall determine 
the total reactions due to load and weight directly'. 

The beam being 20 feet long, its weight is 42 X 20, or 840 
pounds. Since the middle of the beam is 8 feet from the left and 
6 feet from the right su])port, the moments of the weiolit witli 
respect to the left and right supjjorts are respectively: 

840X8 :-- 0,720, and— S4()xr) -^ —5,040 foot-pounds. 

The moment equations foi- all the forces "applied to tlu» beam 
for origins at V> and 1) are like those in exampK' 2, with an addi- 
tional term, the moment of the weight; they are res])ecti\eiy : 
— 2,100 X 2 + -j-3,()0()xO—R,X 14-M,()00xl8 + (>,720-::0, 
—2,100 X 10 + li. X 1 1— 3,(K)() X 8 + 0+ 1,()()0 X 4—5,040 -- 0. 



24 



STEEIs^GTII OF MATERIALS 



•Tlie first equation reduces to 

U R=-52,920, or E,=3,780 pounds, 
and the second to 

14 R,= 61,040, or 1\= 4,360 pounds. 

The sum of the loads and weight of beam is 8,140 pounds; 
and since the sum of the reactions is the same, the computation 
checks. 

EXAMPLES FOR PRACTICE. 

1. AB (Fig. 11) represents a simple beam su23ported at its 
ends. Compute the reactions, neglecting the weight of the beam, 

. j Right reaction = 1,443.75 pounds. 
I Left reaction = 1,556.25 pounds. 



eoolbs. 



k— 2'- 



Qoolbs. soolbs. 

3«-l'-^< 



1000 lbs. 



Fig. 11. 

2. Solve example 1 taking into account the weight of the 

beam, whicli suppose to be 400 pounds. 

. \ Right reaction = 1,043.75 pounds. 
( Left reaction = 1,756.25 pounds. 

3. Fig. 12 represents a simple beam weighing 800 pounds 

supported at A and B, and sustaining three loads as shown. 

What are the reactions ? 

i j Right reaction = 2,014.28 pounds. 
( Left reaction = 4,785.72 pounds. 



20 


oo lbs. 


cl 


xl 


looolbs- 


n' 


30OO 
J 


lbs. 


K-1 -^ 

1 > 


i L)— -■ ^ 


p- 2 '^ 


y 












i 


'A 






B" 



Fig. 12. 

4. Suppose that in example 3 the beam also sustains a uni- 
formly distributed load (as a floor) over its entire length, of 500 
pounds per foot. Compute the reactions due to all the loads and 



the weight of the beam. 



Ans. 



Right reaction == 4,871.43 pounds. 
Left reaction = 11,928.57 pounds. 



STRENGTH OF MATERIALS 



EXTERNAL SHEAR AND BENDING MOMENT. 

On almost every cross-section of a loaded beam there are 
three kinds of stress, namely tension, compression and shear. The 
first two are often called jihre stresses because they act along the 
real fibres of a wooden beam or the imaginary ones of which we 
may suppose iron and steel beams composed. Before taking up 
the subject of these stresses in beams it is desirable to study certain 
quantities relating to the loads, and on which the stresses in a 
beam depend. These quantities are called external shear and 
bending moment, and will now be discussed. 

34. External Shear. By external shear at (or for) any sec- 
tion of a loaded beam is meant the algrebraic sum of all the loads 
(including weight of beam) and reactions on either sale of the 
section. This sum is called external shear because, as is shown 
later, it equals the shearing stress (internal) at the section. For 
brevity, we shall often say simply ''shear" when external shear is 
meant. 

35. Rule of Signs. In computing external shears, it is cus- 
tomary to give the plus sign to the reactions and the minus sign 
to the loads. But in order to get the same sign for the external 
shear whether computed from the right or left, we chaiuje the sujii 
of the sum w^hen computed from the loads and reactions to the 
rujht. Thus for section a of the beam in Fig. 8 the algebraic sum is, 
when computed from the left, 

-1,000 + 3,000 =-3 +2,000 pounds; 
and when computed from the right, 

-1,000 + 3,000-2,000-2,000 .-. -2,000 pounds. 
The external shear at section a is +2,000 ])ounds. 

Again, for section 1> the algebraic sum is, 
when computed from the left, 

-1,000 + 3,000-2,000-2,000 + 8,000 = + 1,000 pounds; 
and when computed from the right, -1,000 jiounds. 

The external shear at the section is +1,000 pounds. 

It is usually convenient to compute the shear at a section 
from the forces to the riolit or left nccordino' as tlu're are fewer 
forces (loads and reactions) on the I'ight or left sides of the 
section. 



26 stee:n^gth of mateeials 

36. Units for Shears. It is customary to express external 
shears in pounds, but any other unit for expressing force and 
weight (as the ton) may be used. 

37. Notation. AYe shall use Y to stand for external shear at 
any section, and the shear at a particular section will be denoted 
by that letter subscripted; thus Y^, Y.^, etc., stand for the shears 
at sections one, two, etc., feet from the left end of a beam. 

The shear has different values Just to the left and right of a 
support or concentrated load. AYe shall denote such values by Y' 
and Y"; thus Y.' and Y." denote the values of the shear at sec- 
tions a little less and a little more than 5 feet from the left end 
respectively. 

Examjples. 1. Compute the shears for sections one foot 
apart in the beam represented in Fig. 9, neglecting the weight of 
the beam. (The right and left reactions are 3,700 and 2,300 
pounds respectively; see example 1, Art. 33.) 

All the following values of the shear are computed from the 
left. The shear just to the right of the left support is denoted by 
Y,/', and Y^" = 2,300 pounds. The shear just to the left of B is 
denoted by Y/, and since the only force to the left of the section 
is the left reaction, Y/ = 2,300 pounds. The shear just to the 
right of B is denoted by Y/', and since the only forces to the left 
of this section are the left reaction and the 1,000-pound load, 
Y/' = 2,300 - 1,000 = 1,300 pounds. To the left of all sections 
between B and C, there are but two forces, the left reaction and 
the 1,000-pound load; hence the shear at any of those sections 
equals 2,300 - 1,000 == 1,300 pounds, or 

Y, = Y3 = Y, = Y, = Y;= 1,300 pounds. 
The shear just to the right of C is denoted by Y,/'; and since the 
forces to the left of that section are the left reaction and the 
1,000- and 2,000-pound loads, 

Y;' == 2,300-1.000-2,000 =- 700 pounds. 

Without further explanation, the student should understand 
that 

Y, — + 2,300 - 1,000 - 2,000 =. - 700 pounds, 

v; =-700, 

y}' = -f 2,300 - 1,000 - 2,000 - 3,000 = - 3,700, 

V, =Y.;=- 3,700, 

Y,;- = ^ 2, 300 - 1,000 - 2,000 - 3,000 H- 3,700 = 



STRENGTH OF MATEPJALS 27 

2. A simple beam 10 feet long, and supported at each end, 
weighs 400 pounds, and bears a uniformly distributed load of 
1,600 pounds. Compute the shears for sections two feet apart. 

Evidently each reaction equals one-half the sum of the load 
and weight of the beam, that is, -J (1,600-^400) =1,000 pounds. 
To the left of a section 2 feet from the left end, the forces actin^j 
on the beam consist of the left reaction, the load on that part of 
the beam, and the weight of that part ; then since the load and 
weight of the beam j[)er foot equal 200 pounds, 

Y,= 1,000-200 X 2 = 600 pounds. 
To the left of a section four feet from the left end, the forces 
are the left reaction, the load on that part of the beam, and the 
weight ; hence 

Y,= 1,000-200 X 4 = 200 pounds. 
Without further explanation, the student should see that 

V, = 1,000-200 X 6 =-200 pounds, 
Y3 = 1,000-200 X 8 = -600 pounds, 
Y^; -- 1,000-200 X 10 = -1,000 pounds, 
Y,;'-- 1,000-200x10+1,000 = 0. 

3. Compute the values of the shear in example 1, taking 
into account the weight of the beam (400 ])Ounds). (The right 
and left reactions are then 3,900 and 2,500 pounds respectively; 
see example 3, Art. 33.) 

AVe proceed just as in example 1, exce])t that in each compu- 
tation we include the weight of the beam to the left of the section 
(or to the right when computing from forces to the right). The 
weiglit of the beam being 40 j)ounds ])er foot, then (^computing 
from the left) 

Y;' ^+2,500 pounds, 

Y; =+2,500-40 ^ + 2,460, 

Y/' = + 2,500-40-1,000-- + 1,46)0, 

Y, = + 2,500-1,000-40x2-- +1,420, 

Y, = f 2,500-1,000-40 X 3 = + l,3cS0, 

Y, = + 2,500-1,000-40x4 = +1,340, 

Y,. =+2,500-1,000-40x5 = +1,300, 

Y; = + 2,500-1,000-40x6= +1,26)0, 

Y./' -H 2,500-1,000-40 x 6-2,000 - 740, 

V, =-1 2,500 1,000 2,000 10x7- 7M), 



28 STKEXGTH OF MATEEIALS 



'J 



Y; = + 2,500-1,000-2,000-40x8 =-820, 
Y;' = + 2,500-1,000-2,000-40x8-3,000 =-3,820, 
Yg = + 2,500-1,000-2,000-3,000-40 X 9 = -3,860, 
Y'lo == + 2,500-1,000-2,000-3,000-40 X 10 = -3,900, 
^^"io= + 2,500-1,000-2,000-3,000-40x10 + 3,900=0. 

Computing from tlie right, we find, as before, that 

Y, =- ( 3,900-3,000-40 X 3) =-780 pounds, 
Y; =-( 3,900-3,000-40 x2)=-820, 
Y;' =-(3,900-40 x2)=^3,820, 

etc., etc. 

EXAMPLES FOR PRACTICE. 

1. Compute the values of the shear for sections of the beam 
represented in Fig. 10, neglecting the weight of the beam. (The 
right and left reactions are 3,300 and 4,000 pounds respectively; 
see example 2, Art. 33.) 

Y, =Y;=-2,100 pounds, 

Y," =Y3= Y,= Y = Y,= Y,= Y;= + 1,900, 

Y " = Y =Y =Y =Y =Y ==Y =Y = Y' =-1 TOO 

'^ 8 '^ 9 * 10 '^ U '^ 12 '13 * U '^ 15 * 16 "^5 ' ^^5 

Y" =Y =Y =Y =Y' = + 1600 

* 16 '^ 17 * 18 * 19 * 20 ^J.jV7VV7. 

2. Solve the preceding example, taking into account the 
weight of the beam, 42 pounds per foot. (The right and left 
reactions are 3,780 and 4,360 pounds respectively; see example 4, 
Art. 33.) 

Y;' = - 2,100 lbs. Y. = + 1,966 lbs. Y,, = - 1,928 lbs. 
Y, =-2,142 Y; = + 1,924 Y,, =- 1,970 



Ans. ^ 



Ans. 



Y ' 

^ 2 


= - 2,184 


y: 


=- - 1,676 


Y ' 

^ 16 


= -2,012 


V/' 


= + 2,176 


y. 


=:- 1,718 


y.r 


= + 1,768 


y. 


= + 2,134 


y.o 


= - 1,760 


Vn 


= + 1,726 


V. 


= + 2,092 


v„ 


= -1,802 


y. 


= + 1,684 


y. 


= + 2,050 


^12 


= - 1,844 


y. 


= + 1,642 


y. 


= + 2,008 


^13 


= -1,886 


^.' 


= + 1,600 



3. Compute the values of the shear at sections one foot apart 
in the beam of Fig. 11, neglecting the weight. (The right and 
left reactions are 1,444 and 1,556 pounds respectively; see example 
1, Art. 33.) 



STEEKGTH OF MATEKIALS 



29 



Ans. 



Y^ =~ 4,160 
Y, =- 4,700 

y; = - 5,240 

y;'= +6,690 



y ' 

17 


— = 


- 250 


V,/' 


: — : 


-3,250 


V. 


= 


- 3,790 


y^. 


= 


- 4,330 


Y ' 

' 20 


= 


- 4,870 


y " 

* 20 









fy;' =y^r=r:y;=+ 1,556 pounds, 
! v;' =y3==y^=y =y;=+956, 
Aiib.-^ y;' =y/=:+56, 

Y " =y =ry =y =y =y =y '=-444 

7 '^8 ^ '■) * 10 ^11 * 12 '13 -i^-t-ty 

, Vi3"=V,«=V,5=V,6'=-l,M4. 

4. Compute the vertical shear at sections one foot apart in 
the beam of Fig. 12, taking into account the weight of the beam, 
800 pounds, and a distributed load of 500 pounds per foot. (The 
right and left reactions are 4,870 and 11,930 pounds respectively; 
see examples 3 and 4, Art. 33.) 

^ y„ = y, = + 6,150 lbs. y,, =+ 830 Ibs. 

V/ = _ 540 lbs. yV = + 5,610 yj -=+ 290 
y/'= _ 2,540 y;' = + 4,610 

y, = - 3,080 y, = + 4,070 

Y^ =~ 3,620 y^o = + 3,530 

y„ =+2,990 

y^^ = +2,450 

y,3 = + 1,910 

y,,= + 1,370 

38. Shear Diagrams. The way in which the external shear 
varies from section to section in a beam can be well represented 
by means of a diagram called a shear diagram. To construct 
such a diagram for any loaded beam, 

1. Lay off a line equal (by some scale) to the length of 
the beam, and mark the positions of the supports and the loads. 
(This is called a "base-line.") 

2. Draw a line such that the distance of any point of it 
from the base equals (by some scale) the shear at the correspond- 
ing section of the beam, and so that the line is above the base 
where the shear is })Ositive, and below it where negative. (This is 
called a Khctcr Ihie^ and the distance from a point of it to the 
base is called the "ordinate" from the base to the shear liiu^ at 
that j)()iiit.) 

AV^e shall (explain these diagrams furthcM' by means of illiis- 
trative examj)les. 

Kxdin plt'x. 1. It is iHMpiired to constcnct llu' shear iliajj;rani 
for tht^ beam represented in Fig. 13,^^ (^a c()j)y of Fig. 9). 



30 



STREXGTH OF MATEEIALS 



Lay off A'E' (Fig. 13. h ) to represent the beam, and mark the 
positions of the loads B', C and D'. In example 1, Art. 37, we 
computed the values of the shear at sections one foot apart; hence 
we lay off ordinates at points on A'E' one foot apart, to represent 
those shears. 

Use a scale of 4,000 pounds to one inch. Since the shear for 
any section in AB is 2.300 pounds, we draw a line o^j parallel 
to the base 0.575 inch (2,300^-4.000) therefrom; this is the shear 
line for the portion AB. Since the shear for any section in BC 
equals 1.30O pounds, we draw a line V c parallel to the base and 



le-i' 



looolbs. 



T 



2000 lbs. 3000 lbs. 



1 



D' 



c' 



ScaJe: i" =4-000 lbs. 



Fiff. 13. 



d' 



E' 



0,325 inch ('1,300-^4.000') therefrom; this is the shear line for the 
portion BC. Since the shear for any section in CD is -700 
pounds, we draw a line c'd below the base and 0.175 inch 
(700-^4,000) therefrom; this is the shear line for the portion 
CD. Since the shear for any section in DE equals -3.700 lbs., we 
draw a YuiQcTe below the base and 0.925 inch (3,700-^4,000) there- 
from; this is the shear line for the portion DE. Fig. 13. />. is the 
required shear diagram. 

2. It is required to construct the shear diao-ram for the 
beam of Fig. 14, <x (a copy of Fig. 9), taking into account the 
weight of the bea-m, 400 pounds. 

The values of the shear for sections one foot apart were com- 
puted in example 3, Art. 37, so we have only to erect ordinates at 
the various points on a base line A'E' (Fig. 14. ^> ), equal to those 



STRENGTH OF MATERIALS 



31 



values. We shall use the same scale as in the preceding illustra- 
tion, -itjOOO pounds to an inch. Then the lengths of the ordi nates 
corresponding to the values of the shear (see example 3, Art. 37) 
are respectively: 

2,500^4,000=0.625 inch 

2,460-^^4,000=0.615 " 

1,460-^4,000=0.365 " 
etc. etc. 

Laying these ordinates off from the base (upwards or downwards 
according as they correspond to positive or negative shears), we 
get al)^ JjG^ cd^ and d'e as the shear lines. 

loooltoc. aooolbs. 3oooltas. 




Scex.le i"=4-ooolb5. 



Pig. 14. 
3. It is required to construct the shear diagram for the 
cantilever beam represented in Fig. 15, «, neglecting the weight 
of tlie beam. 

Tlie value of the shear for any section in AB is -500 ])ounds; 
for any section in 1>(\ -1,500 pounds; and for any section in 
01)^ _;-^^r)00 pounds. Hence the shear lines are ah, b'c, cd. The 
scale being 5,000 pounds to an inch, 

AV. = 500--5,000 ^-. 0.1 inch, 
]]'// = 1,500^5,000 -^(1.3 - 
V/c' = 3,500-^5,000 = 0.7 '* 
riui shear lines wvv all below the base because all the values of the 
shear are neiiative. 



32 



STEENGTII OF MATEKIALS 



4. Suppose that the cantilever of the preceding illustration 
sustains also a uniform load of 200 pounds per foot (see Fig. 16, a). 
Construct a shear diaofrani. 



Soolbs. looolbs. 2ooolbs. 
L — 2i — L 3» ^ ^ 



A 



IB 
I 



t 



I 



-.1 a. 



a- 






c' 
Scawle i"=5ooolbs. 

Fig. 15. 

First, we compute the values of the shear at several sections. 
Thus y;' ==.- 500 pounds, 

Y, ==- 500 - 200= - 700, 
Y; =-500 - 200x2=-900, 
Y;' =~ 500 - 200X2 - 1,000=-1,900, 
Y^ ^_ 500 _ 1,000 - 200 X 3=-2,100, 
Y, =- 500 - 1,000 - 200 X l=-2,300, 
Y/ =- 500 - 1,000 - 200 X 5=-2,500, 
Y/' =- 500 - 1,000 - 200x 5 - 2,000^-4500, 
=- 500 - 1,000 - 2,000 - 200x6=-l,700, 
=~ 500 - 1,000 - 2,000 - 200xT=:-l,900, 
:=_ 500 - 1,000 - 2,000 - 200X8=:-5,100, 
r=. _ 500 - 1,000 - 2,000 - 200 X 9=-5,300. 
The values, being negative, should be plotted downward. To a 
scale of 5,000 pounds to the inch they give the shear lines ah^ h'c^ 
c'd (Fig. 16, h), 

EXAMPLES FOR PRACTICE. 

1. Construct a shear diagram for the beam represented in 
Fig. 10, neglecting the weight of the beam (see example 1, Art. 37). 

2. Construct the shear diagram for the beam represented in 
Fig. 11, neglecting the weight of the beam (see example 3, 
Art. 37). 






STEEXGTII OF MATEEIALS 



33 



3. Construct the shear diagram for the beam of Fig. 12 
when it sustains, in addition to the loads represented, its own 
weight, 800 pounds, and a uniform load of 500 pounds per foot 
(see example 4, Art. 37). 

4. Figs, a^ cases 1 and 2, Table B (page 55), represent two 
cantilever beams, the first bearing a concentrated load P at the free 
end, and the second a unifol-m load W. Figs, h are the corre- 
sponding shear diagrams. Take P and W equal to 1,000 pounds, 
and satisfy yourself that the diagrams are correct. 

5. Figs. «, cases 3 and 4, same table, represent simple 
beams supported at their ends, the first bearing a concentrated 



50olbs. looolbe. aooolbs. 




Sc5-le !"=« 5000 lbs. 
Fig. 16. 

load P at the wiiddle, and the second a uniform load W. Fitrs. 
h are the corresponding shear diagrams. Take P and W ec^ual 
to 1,000 pounds, and satisfy yourself that they are correct. 

39. Maximum Shear. It is sometimes desirable to know 
the greatest or maximum value of the shear in a given case. This 
value can always be found witli certainty by constructing the shear 
diacrram, from which the maximum value of the shear is evident at 
a glance. In any case it can most readily be computed if oiu* 
knows the section for which the shear is a maximum. The stu- 
dent should examine all the shear diat>*rams in the i)recedini'" 
articles and those that he has drawn, and see that 

1. hi cimtili'oei's Jid'cd in a tcdll^ thi' nma^ujiujn tilnar 
occurs (it t/io wulL 



34 ' STRENGTH OF MATERIALS 



2. In Hunple heanis, the maxiimmi shear occurs at a sec- 
tion next to one of the supports. 

By the use of these propositions one can determine the value 
of the maximum shear without constructing the whole shear 
diagram. Thus, it is easily seen (referring to the diagrams, page 
55) that for a 

Cantilever, end load P, maximum shear=P 

" , uniform load W, " " =AV 

Simple beam, middle load P, " '» =4^ 

" " , uniform " AY, '' '• =1W 

40. Bending floment. By bending moment at (or for) a 
section of a loaded beam, is meant the algebraic sum of the mo- 
ments of all the loads (including weight of beam) and reactions 
to the left or right of the section with respect to any point in the 
section. 

41. Rule of Signs. A¥e follow the rule of signs previously 
stated (Art. 29) that the moment of a force which tends to pro- 
duce clockwise rotation is plus, and that of a force which tends to 
produce counter-clockwise rotation is minus; but in order to get 
the same sign for the bending moment whether computed from 
the right or left, we change tJie sign of the sum of the moments 
when computed from the loads and reactions on the right. Thus 
for section <2, Fig. 8, the algebraic sums of the moments of the 
forces are: 

when computed from tne left, 

-1,000 X 5 + 3,000 X 1 =-2,000 foot-pounds ; 
and when computed from the right, 

1,000 X 19-3,000 X 15 + 2,000 X 13 + 2,000 X 1 = -f 2,000 foot- 
pounds. 
The bending moment at section a is -2,000 foot-pounds. 

Again, for section J, the algebraic sums of the moments of the 
forces are: 
when computed from the left, 

-1,000 X 22 + 3,000 X 18-2,000 X 16-2,000 X 1 + 3,000 X 2= 
-2,000 foot-pounds; 
and when computed from the right, 

1,000 X 2= + 2,000 foot-pounds. 
The bending moment at the section is -2.000 foot-DOunds. 




CO 




en 




U 




K 




H 




t/l 






s>> 


C5 




Z 


"i; 


^^ 


•^ 


H 




CO 

3 


^ 


OC 


-^ 


X 


^ 


H 


•_^ 






CC 


^ 


(d 


-2 


Q 


. 


Z 


^ 


3 


"2 






Vi 


V* 


Q 


^ 


O 


W 


o 


^ 


^ 


■-. 


C/5 


,. 


3 


.;^ 


O 


"^- 






o: 


^ 


< 


c 






> 


--^ 


(X. 


•:- 


O 


«i 






b] 


^ 


OC 


X 


Q 


^ 


F- 


'i. 


U 


^ 


< 


j; 


CC 




b. 


5 










O 


V 


z 


S 


►< 


X 


^ 


^ 
w 


o 


•3 


X 


o 


(f) 


I.. 




ft. 


z 




mm 




^ 




< 




K 




Q 





STEEXGTTI OF MATEEIALS 35 

It is usually convenient to compute the bending moment for 
a section from the forces to the riorht or left accordincr as there 
are fewer forces (loads and reactions) on the right or left side 
of the section. 

42. Units. It is customary to express bending moments in 
inch-pounds, but often the foot-pound unit is more convenient. 
To reduce foot -jxjunds to inch-jpounds^ midtijfljj l>]j ticelve. 

43. Notation. We shall use M to denote bendincr moment at 
any section, and the bending moment at a particular section will 
be denoted by that letter subscripted; thus Mj, M,„ etc., denote 
values of the bending moment for sections one, two, etc., feet 
from the left end of the beam. 

Examples. 1. Compute the bending moments for sections 
one foot apart in the beam represented in Fig. 9, neglecting the 
weight of the beam. (The right and left reactions are 3,700 and 
2,300 pounds respectively. See example 1, Art. 33.) 

Since there are no forces acting; on the beam to the left of the 
right support, ]\I„=0. To the left of the section one foot from the 
left end there is but one force, the left reaction, and its arm is one 
foot; hence M, = + 2,300x1 ==2,300 foot-pounds. To the left of 
a section two feet from the left end there are two forces, 2,300 and 
1,000 pounds, and their arms are 2 feet and 1 foot respectively; 
hence M,= + 2,300 X 2-1,000 X 1=3,600 foot-])ounds. At the 
left of all sections between B and C there are only two forces, 
2,300 and 1,000 pounds; hence 

M, = -h 2,300 X 3-1,000 X 2=- -|- 4,900 foot-])ounds, 
]VI,= -k 2,300 X 4-1,000 X 3= + 6,200 

M,.= + 2,300 X 5-1,000 X 4= + 7,r)00 
M^ = -h 2,300 X ()-l,000x 5= + 8,800 

To the rio;ht of a section sev^en feet from the left end tliere 
are two forces, the 3,()00-j)oun(l h)a(l aiid the right reaction 
(3,700 pounds), and their arms with respect to an origin in tliat 
section are respectively one foot and three feet; hence 

M,=-(-3,70()X3 [ 3,00()Xl) = -f-8,100 foot-pounds. 

To tlie riirht of any section between K and 1> tiiere is only one 

O t/ » 

force, the right reaction ; hence 



36 STKElSrGTH OF MATERIALS 

M,=-(-3,700x 2)=7,400 foot-pounds, 

M,=--(-3,700 X l)=-3,700 

Clearly M,,=0. 

2. A simple beam 10 feet long and supported at its ends 
weighs 400 pounds, and bears a uniformly distributed load of 1,600 
pounds. Compute tlie bending moments for sections two feet 
apart. 

Each reaction equals one-half the whole load, that is, ^ of 
(1,600 + 4:00) =1,000 pounds, and the load per foot including 
w^eight of the beam is 200 pounds. The forces acting on the 
beam to the left of the first section, two feet from the left end, are 
the left reaction (1,000 pounds) and the load (including weight) 
on the part of the beam to the left of the section (400 pounds). 
The arm of the reaction is 2 feet and that of the 400 -pound force 
is 1 foot (the distance from the middle of the 400-pound load to 
the section). Hence 

M,= + 1,000 X 2-400 X 1 = + 1,600 foot-pounds. 

The forces to the left of the next section, 4 feet from the left 
end, are the left reaction and all the load (including w^eight of 
beam) to the left (800 pounds). The arm of the reaction is 4 feet, 
and that of the 800-pound force is 2 feet; hence 

M,= + 1,000 X 4-800 X 2 = -f 2,400 foot-pounds. 

Without further explanation the student should see that 

M^ = + 1,000 X 6-1,200 X 3= + 2,400 foot-pounds, 
M^= +1,000 X 8-1,600 X 4= + 1,600 

Evidently M,==Mjo=0. 

3. Compute the values of the bending moment in example 
1, taking into account the weight of the beam, 400 pounds. (The 
right and left reactions are respectively 3,900 and 2,500 pounds; 
see example 3, Art. 33.) 

We proceed as in example 1, except that the moment 
of the weight of the beam to the left of each section (or to 
the right when computing from forces to the right) must be 
included in the respective moment equations. Thus, computing 
from the left. 



STRENGTH OF MATEPJALS 



37 



Ml = + 2,500 X 1-40 X -I = + 2,480 foot-pounds, 
M^ = + 2,500 X 2-1,000 X 1-80 X 1= -f 3,920, 
M^^ = + 2,500 X 3-1,000 X 2-120 X 1^-= -f 5,320, 
M, = + 2,500 X 4-1,000 X 3-160 X 2 = -f 6,680, 
M, = + 2,500 X 5-1,000 X 4-200 X 2i= + 8,000, 
M^ -= + 2,500 X 6-1,000 X 5-240 X 3= 4- 9,280. 

Computing from the right, 

M, =-(-3,900x 3 + 3,000 X 1-4-120 Xli) =-+8,520, 
-(-3,900x2 + 80xl) = + 7,720, '^' 
-(-3,900x1 + 40x4)=+ 3,880, 



M, 
M. 



0. 



EXAMPLES FOR PRACTICE. 



1. Compute the values of the bending moment for sections 
one foot apart, beginning one foot from the k^ft end of the 
beam represented in Fig. 10, neglecting the weight of the beam. 
(The right and left reactions are 3,300 and 4,000 ])ounds respec- 
tively; see example 2, Art. 33.) 

f M,-= - 2,100 M, :.- + 3,400 M„=:: + 2,100 M,,=-6,400 

Ans. I M.,= -4,200 M. -= + 5,300 M„=+ 400 M,,=-4,.S(H) 

(in foot- J M.= - 2,300 M^ = + 7,200 M,l= - 1,300 M„---3,200 

pounds) M = - 400 M„ === + 5,500 M,,-= - 3,000 M„=:-l,600 

[ M,== + 1,500 M, = + 3,800 M,^^- - 4,700 U,,= 

2. Solve the preceding example, taking into account the 
.weight of the beam, 42 pounds per foot. (The riglit and left 
reactions are 3,780 and 4,360 pounds respectively; see exampK' 4, 
Art. 33.) 

"M,=: - 2,121 M,, =+4,084 ]M„ = + 2,799 M„= - 6,7;^(] 

Ans. M,= - 4,284 JVL = + 6,071 M,,= + 976 M,,= -4,9S9 

(in foot- \ M. = - 2,129 M^ =+8,016 iAI,,-- - 8S9 iAI„= - 3,2s4 

P^^""<^«) M*= - 1(5 M,, = + 6,319 JVl,,-- - 2,796 U,,-^ - 1,621 

M,= + 2,055 Mi„= + 4,580 M„= - 4,745 M.„= 

3. Compute the bending moments for scH'tions one foot 
a])art, of the beam represented in Kig. 11, iicgUvting tlie weiglit. 
(The right and left reactions are 1. 1 I I and \J)T)i\ pounds respect- 
ively; see exampU^ 1, Art. 3)^) 



88 



STEEKGTH OF MATEEIALS 



M = + 1,556 ]\L = + 5,980 M, = + 6,104 M,3= + 4:,328 
(in foot 4 ^^.-- + 3,112 M,= + 6,936 M,= + 5,660 M,= + 2,884 
pounds) I ^^3- + 4068 ]VL= + 6,992 M,= + 5,216 M,= + 1,M0 

Ul,= + 5,024 M,= + 6,548 M,,= + 4,772 M,^^ 

4 Compute the bending moments at sections one foot apart 
in the beam of Fig. 12, taking into account the weight of the beam, 
800 pounds, and a uniform load of 500 pounds per foot. (The 
right and left reactions are 4,870 and 11,930 pounds respectively; 
see Exs. 3 and 4, Art. 33.) 

"M =- 270 M, =-19,720 M,,=+ 3,980 M,,^12,180 

Ans. M,= - 3,080 M, = -13,300 M,,= + 6,700 M,,=12,200 

(in foot- -| M3=- 6,430 M3 =- 7,420 Mj3=+ 8,880 M,3=- 8,680 

pounds) j M,= -10,320 M, =- 3,080 M,,= + 10,520 M,,^ 4,620 

[ ]VL= -14,750 M,,= + 720 M,,= + 11,620 M,,= 

44. Moment Diagrams. The way in which the bendino- 
moment varies from section to section in a loaded beam can be 
well represented by means of a diagram called a vioment (lvifjva::i. 
To construct such a diagram for any loaded beam. 



1000 lbs. 



2000 lbs. 3000 lbs. 




ScaJe:i = looooft.-lbs 

Fig. 17. 

1. Lay off a base-line just as for a shear diagram (see 
Art. 38). 

2. Draw a line such that the distance from any point of it 
to the base-line equals (by some scale) the value of the bending 
moment at the corresponding section of the beam, and so that the 
line is above the base w^here the bending moment is positive and 
below it where it is negative. (This line is called a "moment 
line.") 



STKEIS^GTH OF MATEEIALS 



30 



Examples. 1. It is required to construct a moment dia- 
gram for the beam of Fig. 17, a (a copy of Fig. 0), loaded as 
tliere shown. 

Layoff A'E' (Fig. 17, h) as a base. In example 1, Art. 43, 
we computed the values of the bending moment for sections one 
foot apart, so we erect ordinates at points of A'E' one foot apart, 
to represent the bending moments. 

We shall use a scale of 10,000 foot-pounds to the inch; then 
'he ordinates (see example 1, Art. 43, for values of M) will be: 
One foot from left end, 2,300^10,000 = 0.23 inch, 
Two feet " '' " 3,600-^10,000 -= 0.36 " 
Three" '' '' " 4,900^10,000 = 0.49 " 
Four " " " '' 6,200-f-10,000 = 0.62 " 

etc., etc. 



„ 1 


io< 


50lb9. 


=' 


200C 

-1 


lbs. 


-2 


3000 


lbs 


.«! 


,f 


. 


; 






1 


A ' ., 1 


-^A 




B 




iC 
1 






D 
















c 






i 


Sv 






rTTTTgfllJlll'' 








Tr 




Iffiil! 


tllf 


5rTtTlinfII 



















Ik 



A 



C' D' 

Sce>s.le: i"=» looooft.-lbs. 
Fiff. 18. 



E' 



Laying these ordinates off, and joining their ends in succession, 
we get the line A'hcdK, which is the bending moiniMit line. 
Fig. 17, />, is the moment diagram. 

2. It is required to construct the monu^nt diagram for tlu» 
beam. Fig. 18, a (a copy of Fig. 9), taking into account tlio weight 
of the beam, 400 pounds. 

The values of the bending monuMit for sections one foot apart 
were computed in example 3, Art. 43. So we liave only to lay off' 
ordinates equal to those values, one foot apart, on the base A'E" 
(Fig. 18, I). 

To a scale of 10,()()() foot-j)()unils to tlu' inrli tlii' ordinates 
(see example )i, Art. 43, for values of J\I) an': 



40 



STEEXGTH OF MATERIALS 



At left end, 

One foot from left end, 2,^:80-^10,000=0.248 inch 
T^Yo feet " " " 3,920--10.000=0.392 " 
Three" " ." " 5,320-^10,000=0.532 " 
Four " '' '- " 6,680^10,000=0.668 " 

Laying these ordinates off at the proper points, we get A'JcrZE 

as the moment line. 

3. It is required to construct the moment diagram for the 

cantilever beam represented in Fig;. 19, <r^ neslectincr tue weio-ht 

of the beam. The bending moment at B equals 

-500x2=-l,000 foot-pounds; 
at C, 

-500 X 5-1,000 X 3=-5,500 ; 
and at D, 

-500 X 9-l,000x 7-2,000 X 4^-19,500. 



soolbs. looolbs, 2000 lbs. 




d^ 



ScaJe:i"= 20000 ft.-lbs 



Fig. 19. 

Using a scale of 20,000 foot-pounds to one inch, the ordinates 
in the bendino* moment diagram are: 

" "" AtB, 1,000-f- 20,000=0.05 inch, 
'' 0, 5,500-^20,000=0.275 - 
'• D, 19,500-^20,000=0.975 " 
Hence we lay these ordinates off, and downward because the bend- 
ing moments are negative, thus fixing the points b, <■ and d. The 
bendino; moment at A is zero; hence the moment line connects A 
^, e and d. Further, the portions A^, he and cd are straight, as 
can be shown by computing values of the bending moment for 
sections in AB, BC and CD, and laying off the corresponding 
ordinates in the moment diao-ram. 



STEENGTH OF MATEEIALS 



41 



4. Suppose tliat the cantilever of the precedincr illustration 
sustains also a uniform load of 100 pounds per foot fsee Ficr. 20, a). 
Construct a moment diacrram. 

First, we compute the values of the bending moment at sev- 
eral sections; thus, 

M^=:-500 X 1-100 X i=-550 foot-pounds, 
M^^-SOO X 2-200 X 1 =-1,200, 
M3=-500 X 3-1,000 X 1-300 X li=-2,950, 
M^=-500 X 4-1,000 X 2-400 X 2=-4,800, 
M, =-500 X 5-1,000 X 3-500 X 2i=.-6,750, 
m[=-500 X 6-1,000 X 4-2,000 X 1-600 X 3=-10,800, 
M^=-500 X 7-1,000 X 5-2,000 X 2-700 X 3^=-14,950, 
M^=-500 X 8-1,000 X 6-2,000 X 3-800 X 4^-19,200, 
M =-500 X 9-1,000 X 7-2,000 X 4-900 X 4* =-23,550. 



soolbs. leoolbs. 



2ooolbs. 



A'-^^ir 




"b 



. Sce>Je:i"=2ooooft.-lb3 



Fig. 20. 

These values all beino^ netrative, the ordinates are jdl laid off 
downwards. To a scale of 20,000 f();)t- pounds to one ineli. tliey 
fix the moment line A'hcd. 

EXAHPLES FOR PRACTICE. 

1. Construct a moment diagr:nii for ibe l>eam representi'd in 
Fio;. 10, neo'h'ctinfy the wei(dit of the bi-nn. (See example I, 
\rt. 43). 

2. Construct a moment diaj^ram I'oi" tlu' brani repfesenlvnl 
in Fio'. 11, ne(ifl(^ctin<'' tlie wciohl of the beam. i Scr example '.\. 
Art. 43). 

3. Construct the moment diagram for the beam of l"'i»:, 12 



42 STEEl^GTH OF MATERIALS 

when it sustains, in addition to the loads represented and its own 
weight (800 pounds), a uniform load of 500 pounds per foot, 
(See example 4, Art. 43.) 

4. Figs. «, cases 1 and 2, page 55, represent two cantilever 
beams, the first bearing a load P at the free end, and the second 
a uniform load W. Figs, h are the corresponding moment 
diagrams. Take P and W equal to 1,000 pounds, and I equal to 
10 feet, and satisfy yourself that the diagrams are correct. 

5. Figs, a^ cases 3 and 4, page 55, represent simple beams 
on end supports, the first bearing a middle load P, and the other a 
uniform load W. Figs, h are the corresponding moment dia- 
grams. Take P and "W equal to 1,000 pounds, and I equal to 
10 feet, and satisfy yourself that the diagrams are correct. 

45. Maximum Bending Moment. It is sometimes desirable 
to know the greatest or maximum value of the bending moment 
in a given case. This value can always be found with certainty 
by constructing the moment diagram, from which the maximum 
value of the bending moment is evident at a glance. But in any 
case, it can be most readily computed if one knows the section for 
which the bending moment is greatest. If the student will com- 
pare the corresponding shear and moment diagrams which have 
been constructed in foregoing articles (Figs. 13 and 17, 14 and 
18, 15 and 19, 16 and 20), and those which he has drawn, he will 
see that — The onaximuin heuding moment in a heaia occurs 
where the shear changes sign. 

By the help of the foregoing principle we can readily com- 
pute the maximum moment in a given case. AVe have only to 
construct the shear line, and observe from it where the shear 
changes sign; then compute the bending moment for that section. 
If a simple beam has one or more overhanging ends, then the shear 
changes sign more than once — twice if there is one overhanging 
end, and three times if two. In such cases we compute the 
bending moment for each section where the shear changes sign; 
the largest of the values of these bending moments is the maxi- 
mum for the beam. 

The section of maximum bendino; moment in a cantilever 
fixed at one end (as when built int3 a wall) is always at the wall. 



STEENGTH OF MATEKIALS 43 

Thus, without reference to the moment diagrams, it is readily seen 
that, 

for a cantilever whose length is I, 

with an end load P, the maximum moment is PZ, 
« a uniform " W, " " " " J MVL 

Also by the principle, it is seen that, 

for a beam whose length is I, on end supports, 

with a middle load P, the maximum moment is J P/, 
" uniform " W, " " " " 1 W^. 

46. Table of Maximum Shears, Moments, etc. Table B 
on page 55 shows the shear and moment diagrams for ei^ht 
simple cases of beams. The first two cases are built-in cantilevers; 
the next four, simple beams on end supports; and the last two, 
restrained beams built in walls at each end. In each case I 
denotes the length. 

CENTER OF GRAVITY AND HOMENT OF INERTIA. 

It will be shown later that the strength of a beam depends 
partly on the form of its cross-section. The following discussion 
relates principally to cross-sections of beams, and the results 
reached (like shear and bending moment) will be made use of 
later in the subject of strength of beams. 

47. Center of Gravity of an Area. The student probably 
knows what is meant by, and how to find, the center of gravity of 
any flat disk, as a piece of tin. Probably his way is to balance 
the piece of tin on a pencil point, the point of the tin at which it so 
balances being the center of gravity. (Ileally it is midway between 
the surfaces of the tin and over the balancing point.) The center 
of gravity of the piece of tin, is also that ])()int oP it tlirongli which 
the resultant force of gravity on the tin (that is, the weight of the 
piece) acts. 

By "center of gravity" of a plane area of any sha|)e we mean 
that point of it wliich corresj)oiHls to tlie center of iii-;i\itv of a 
piece of tin when tlu^ latter is cut out in tlie sli:i|)(> of llio area. 
The center of gravity of a (piite ii'i-cgular area can lie foiiiid nn)st 
rciadily by balancing a piece of tin oi' still' pajHT cut in tlie sliajie 
of the area. But when an area is sinipK' in sliajtc, ov consists of 
parts which are simple, the center of gra\ ity of tlu' whole can l»e 



44 



STKEXGTH OF MATEEIALS 



found readily Dy computation, and such a method will now be 
described. 

48. Principle of rioments Applied to Areas. Let Fig. 21 
represent a piece of tin which has been divided off into any num- 
ber of parts in any way, the weight of the whol.e being AV, and 
that of the parts Wj, W^, ^^, etc. Let Oj, C,, Cg, etc., be the 
centers of gravity of the parts, C that of the whole, and c^, c.,, c^, 
etc., and c the distances from those centers of gravity respectively 
to some line (L L) in the plane 




Ficr. 21. 



of the sheet of tin. 'Wlien the 
tin is lying in a horizontal posi- 
tion, the moment of the weio-ht 
of the entire piece about L L is 
W^, and the moments of the 
parts are Wi^i, Wg^^g, etc. 'Since 
the weight of the whole is the 
resultant of the weights of the 
parts, the moment of the weight 
of the whole equals the sum of the moments of the weio-hts of the 
parts ; that is, 

■Vrc=^V,-fW.A^etc . . . . 

Xow let Aj, A2, etc. denote the areas of the parts of the pieces 
of tin, and A the area of the whole; then since the weio-hts are 
proportional to the areas, we can replace the ^\"'s in the preceding 
equation by corresponding A's, thus: 

Ar=A/'i-L A/.-,-f-etc (4) 

If we call the product of an area and the distance of its 
center of gravity from some line in its plane, the "moment" of the 
area with respect to that line, then the preceding equation may be 
stated in words thus: 

The raoment of an area vnth respect to any line equals the 
algehra'ic sum of the moments of the ])arts of the area. 

If all the centers of gravity are on one side of the line with 
respect to which moments are taken, then all the moments should be 
given the plus sign; but if some centers of gi'avity are on one side 
and some on the other side of the line, then the moments of the 
areas whose centers of gravity are on one side should be giver the 



STEENGTH OF MATERIALS 



45 



same sign, and the moments of the others the opposite sign. The 
foregoing is the principle of moments for areas, and it is the basis 
of all rules for finding the center of gravity of an area. 

To find the center of gravity of an area which can be divided 
np into simple parts, we write the principle in forms of equations 
for two different lines as "axes of moments," and then solve the 
equations for the unknown distances of the center of gravity of the 
whole from the two lines. We explain further Ijy means of specific 
examples. 

Examjples. 1. It is required to find the center of gravity 
of Fig. 22, «, the width being uniformly one incli. 

The area can be divided into two rectano-les. Let C, and 

L" 



?i 



OK- 



_L' 



•C 



J I 



-12' 
"b 



Fig. 22. 

O3 be the centers of gravity of two such parts, and the center of 
gravity of the whole. Also let a and Ij denote the distances of C 
from the two lines OL' and OL" respectively. 

The areas of the parts are 6 and 8 square inches, and their 
arms with respect to OL' are 4 inches and i inch respectivi'h, and 
with resj)ect to OL" -J inch and 1^ inches. Hence the eipnitions of 
moments with respect to 01/ and OL" (the \\hoK' area being 9 
square inches) are: 

9X^^ = 0X4 + 3X^ = 25.5, 

i)X/> = 0X4 + 3X14= 7.5. 
Hence, ' <r 25.5-^0 = 2.S8 inches, 

h : 7.5-^9 ---- 0.83 - . 

2. It is reipiired to hx-atetlu' ccnti'i' of gi-;i\it v of I'ig. '2'-!^ h^ 
the width luMno- uniformly one inch. 



46 



STRENGTH OF MATERIALS 



The ficriire can be divided up into three rectano-les. Let Ci, C, 
and C3 be the centers of gravity of such parts, C the center of 
gravity of the whole; and let a denote the (unknown) distance of 
C from the base. The areas of the parts are 4, 10 and 4 square 
inches, and their '• arms " with respect to 
the base are 2, -| and 2 inches respectively; 
hence the equation of moments with re- 
spect to the base (the entire area being 18 
square inches) is: 

\%y^a =4x2^-10x*-f4x2 = 21. 

Hence, « = 21^-18 = 1.17 inches. 

From the symmetry of the area it is plain 
that the center of gravity is midway be- 
tween the sides. 



EXAMPLE FOR PRACTICE. 

1. Locate the center of crravity 



1 



of 



-4i'- 



Fig. 23. 



Fig. 23. 



Ans. 2.6 inches above the base. 
49. Center of Gravity of Built=up Sections. In Fig. 24 
there are repre'sented cross -sections of various kinds of rolled steel, 
called ''shape steel," which is used extensively in steel construction. 
Manufacturers of this material publish ''handbooks" giving full 
information in regard thereto, among other things, the position of 
the center of g;ravity of each cross section. AVith such a handbook 



1-Toea.m 



Cha^Tjnel n 



Angle 



Z-ba>-r 



T-bawr 



Fig. 21. 

available, it is therefore not necessary actually to compute the posi- 
tion of the center of gravity of any section, as we did in the pre- 
ceding article; but sometimes several shapes are riveted together to 



STEENGTII OF MATERIALS 



47 



make a "built-up" section (see Fig. 25), and then it may be neces- 
sary to compute the position of the center of gravity of the section. 
Examjyle. It is desired to locate the center of gravity of the 
section of a built-up beam represented in Fig. 25. The beam con- 



14"- 



^ 



1 



<VJ 



C 



\L - 

Fig. 25. 

sists of two channels and a plate, the area of the cross-section of a 

channel being 6.03 square inches. 

Evidently the center of gravity of each channel section is 

inches, and that of the plate section is 12]; inches, from the bottom. 

Let c denote the dis- 
tance of the center of 
gravity of the wlioK' 
section from tlie bot- 
tom; tlien since the 
area of the phite section 
is 7 s(]uare inches, and 
that of the whok' sec- 
tion is 11). ()('), 

ii).()r)X<' = o.o3 xo-f 
i\m x<) + 7xi2| = 



.-Jl— 



3"l 

2r 



dw. 




1.66" 



Fig. 2G. 



Hence, 



6'--:15S.ll-ll).()()=.8.8() inches. 
RXAMPLIuS rOR PRACTICR. 

Locati^ tht^ ccnU'r of gravity of ihr luiill iiji sci'tion of 



48 STEENGTH OF MATERIALS 

Fig. 26, a, the area of each "angle" being 5.06 square inches, and 
the center of gravity of each being as shown in Fig. 26, h. 

Ans. Distance from top, 3.08 inches. 

2. Omit the left-hand angle in Fig. 26, a^ and locate the 
center of gravity of the remainder. 

^ ( Distance from top, 3.65 inches, 
"^' I " " left side, 1.10 inches. 

50. rioment of Inertia. If a plane area be divided into an 
infinite number of infinitesimal parts, then the sum of the prod- 
ucts obtained by multiplying the area of each part by the square 
of its distance from some line is called the moment of inertia of the 
area with respect to the line. The line to which the distances are 
measured is called the inertia -axis,' it may be taken anywhere in 
the plane of the area. In the subject of beams (where we have 
sometimes to compute the moment of inertia of the cross-section 
of a beam), the inertia-axis is taken through the center of gravity 
of the section and horizontal. 

An approximate value of the moment of inertia of an area 
can be obtained by dividing the area into small parts (not infini- 
tesimal), and adding the products obtained by multiplying the 
area of each part by the square of the distance from its center to 
the inertia-axis. 

Example. If the -i-ectangle of Fig. 27, «, is divided into 8 
parts as shown, the area of each is one square inch, and the dis- 
tances from the axis to the centers of gravity of the parts are \ 
and 1^ inches. For the four parts lyingr nearest the axis the 
product (area times distance squared) is: 

lX( -2)'^^\\ ^"^ ^0^' ^^ other parts it is 

Hence the approximate value of the moment of inertia of the area 
with respect to the axis, is 

^(i)+i(l)=10- 
If the area is divided into 32 parts, as shown in Figo 27, 5, 
the area of each part is ^ square inch. For the eight of the little 
squares farthest away from the axis, the distance from their centers 
of gravity to the axis is 1| inches; for the next eight it is 1^; 
for the next eight |; and for the remainder ^ inch. Hence an 



STEEXGTII OF MATEPJALS 



49 



approximate value of the moment of inertia of the rectangle with 
respect to the axis is : 

8xiX(lf)^ + 8xiX(liy-!-8xiX(|)^ + 8xiX(i)^=m 

If we divide the rectangle into still smaller j)arts and form 
the products 

(small area) X (distance)", 

and add the products just as we have done, we shall get a larger 
answer than lOJ.' The smaller the parts into which the rec- 
tangle is divided, the larger will be the answer, but it will never 
be larger than 10|. This 10§ is the sum corresponding to a 

division of the rectangle into an 
infinitely large number of parts 
(infinitely small) and it is the 
exact value of the moment of 
^^s_ inertia of the rectanole with re- 
spect to the axis selected. 

There are short methods of 
computing the exact values of the 
moments of inertia of simple fig- 
ures (rectangles, circles, etc.,), 



dcXiS 



« c 


'" . 




e c 


1" 








■' 






/ 


> 






























"^ 














> 


' 







































Fiff. 27. 



but they cannot be given here since they involve the use of difficult 
mathematics. The foregoing method to obtain approximate val- 
ues of moments of inertia is used esj)ecially when the area is quite 
irregular in shape, but it is given here to explain to the student 
the 'meaning of the moment of inertia of an area, lie should 
understand now that the moment of inertia of an area is sim- 
ply a name for such sums as we have just computed. The name 
is not a fittino; one, since the sum has nothincr whatever to do with 
inertia. It was first used in this connection because the sum is 
very similar to certain other sums which had previously been 
called moments of inertia. 

51. Unit of Moment of Inertia. Tiie product (area X dis- 
tance^) is really the product of four lengths, two in (\\('\\ factoi- ; 
and since a monuMit of iiu'i-tia is thes sum of such products, ti 
monuuit of inertia is also the jtroduct of four lenoths. Now th(» 
[)r()duct of two lengths is an area, the pi'oduct i)f threi' is a \ol- 
ume, and the product of four is nioiiieut oi" inertia — unthinKahle in 



8"- 



50 STEEXGTH OF MATEEIALS 

tlie way in wliicli we can think of an area or volume, and there- 
fore the source of much difficulty to the student. The units of 
these quantities (area, volume, and moment of inertia) are respec- 
tively : 

the square inch, square foot, etc., 

" cubic " , cubic '^ " , 
" biquadratic inch, biquadratic foot, etc.; 
but the biquadratic inch is almost exclusively used in this connec- 
tion; that is, the inch is used to compute 
values of moments of inertia, as in the pre- 
ceding illustration. It is often written 
thus: Inchest 

52. Moment of Inertia of a Rectangle. 

^xis i. Let h denote the base of a rectangle, and a 

YicT 23 ^^^ altitude; then by higher mathematics it 

can be shown that the moment of inertia 
of the rectangle with respect to a line through its center of gravity 
and parallel to its base, is yL- Ix/. 

Example. Compute the value of the moment of inertia of 
a rectangle 4x12 inches with respect to a line through its center 
of gravity and parallel to the long side. 

Here ^=12, and « = 4 inches ; hence the moment of inertia 
desired equals 

J^(12X4')==64 inchest 

EXAflPLE FOR PRACTICE. 

1. Compute the moment of inertia of a rectangle 4x12 
inches with respect to a line through its center of gravity and 
parallel to the short side. Ans. 576 inches*. 

53. Reduction Formula. In the previously mentioned 
''handbooks" there can be found tables of moments of inertia of 
all the cross-sections of the kinds and sizes of rolled shapes made. 
The inertia-axes in those tables are always taken through the cen- 
ter of gravity of the section, and usually parallel to some edge of 
the section. Sometimes it is necessary to compute the moment of 
inertia of a "rolled section" with respect to some other axis, and 
if the two axes (that is, the one given in the tables, and the other) 
are parallel, then the desired moment of inertia can be easily com- 
puted from the one given in the tables by the following rule: 



STEEKGTH OF MATEEIALS 



51 



TJie raoinent of inertia of an area vrith resjpect to any axis 
equals the moment of 'inertia vrlth resjject to a jxcrallel axis 
throtcgJi the center of gravity^ plios the product of the area and 
the square of the distance hetween the axes. 

Or. if I denotes the moment of inertia with respect to any axis; 
I^ the moment of inertia with respect to a parallel axis throucrh 
the center of gravity; A the area; and d the c? 'stance between the 
axes, then 



1=1 +AJ^ 



(5) 



eK-xis 




Exa'injple. It is required to compute the moment of inertia 
of a rectangle 2x8 inches w^ith respect to a line parallel to the 
long side and 4 inches from the center of gravity. 

Let I denote the moment of inertia souo-ht, and I, the moment 
of inertia of the rectangle with respect 
to a line parallel to the long side and 
through the center of gravity (see Fig. 
28). Then 

Iq=-^^^«'^ (see Art. 52); and, 

since J=8 inches and ^/ = 2 inches, 

1^=3^2 (8 X 2^) = 5| biquadratic inches. 

The distance between the two inertia- 
axes is 4 inches, and the area of the 
rectangle is 16 square inches, hence 
equation 5 becomes ^^^- '^'^• 

I=5i + 16x4'=261-i biquadratic inches. 

EXAMPLE FOR PRACTICE. 

1. The moment of inertia of an "angle" "2oX2x| inches 
(lengths of sides and width respectively) with respect to a line 
through the center of gravity and parallel to the long side, is 0.()4 
inchest The area of the section is 2 sipiare inches, and tlie dis- 
tance from the center of gravity to tlie long side is 0.1)3 inches. 
(These values are taken from a 'diandhook".) It is rcipiiri'd to 
compute the moment of inertia of the section witli resj)ect to a 
line paralUd to the long sick^ and 4 inches from the center of 
gravity. Ans. ;>2.()4 inches*. 

54. Moment of Inertia of Built-up Sections. As before 
stated, beams are sometinu's ''built ui)"* of roUeil shapi's (angles, 



STEEXGTH OF MATEEIALS 



channels, etc.). The moment of inertia of such a section with 
respect to a definite axis is computed by adding the moments of 
inertia of the parts, all luith respect to that same axis. This is 'the 
method for computing the moment of any area which can be 
divided into simple parts. 

The moment of inertia of an area which may be regarded as 
consisting of a larger area m in us other areas, is computed by sub- 
tracting from the moment of inertia of the large area those of the 
''minus areas." 

ExamjHes. 1. Compute the moment of inertia of the built- 
up section represented in Fig. 30 (in part same as Fig. 25) with 
respect to a horizontal axis 



14' 



SL--^ 






e4-.. 




passing through the center 

of gravity, it being given 

that the moment of inertia 

of each channel section 

with respect to a horizontal ^_ ^ 

axis through its center of 

gravity is 128.1 inches*, 

and its area 6.03 square 

inches. 

The center of gravity of 
the whole section was found 

in the example of Art. 49 to be 8.30 inches from the bottom of 
the section; hence the distances from the inertia-axis to the 
centers of gravity of the channel section and the plate are 2.30 
and 3.95 inches respectively (see Fig. 30). 

The moment of inertia of one channel section with respect to 
the axis A A (see equation 5, Art. 53) is: 

128.1 + 0.03x2.30^=160.00 inchest 
The moment of inertia of the plate section (rectangle) with re- 
spect to the line a" a" (see Art. 52) is: 

_i_ ba=^=3-V[14:X (l);l = 0.15 inches^ 
and with respect to the axis AA (the area being 7 square inches) 

it is: 

015+7x3.95^=109.37 inchest 

Therefore the moment of inertia of the whole section with re- 
spect to AA is: 

2x160.00 + 109.37=3 429.37 inches^. 



STKEIS^GTH OF MATERIALS 



53 



2. It is required to compute the moment of inertia of the 
•'hollow rectangle " of Fig. 29 with respect to a line through the 
center of gravity and parallel to the short side. 

The amount of inertia of the large rectangle with respect to 
the named axis (see Art. 52) is: 



B 



o 



'CO 

o 



dK. 



u 



_C|^ 



iTo 



Ib 

Fi^. 31. 

and the moment of inertia of the smaller one with resj)ect to the 
same axis is: 

iV(4x8')=l'r0f; 

hence the moment of inertia of the hollow section with res])ect 
to the axis is: 

4ir)§ - 170§ = 24f) inchest 

EXAMPLES FOR PRACTICE. 

1. ('ompute the moment of inertia of the section repre- 
sented in Fig. 81, <(^ about the axis AA, it being 8.0^ inches 
from tlie to[). (iliven also tliat the area of one angk^ section is 
5.00 sipiare inches, its center of gravity V (Fig. 31, It) !.()() inches 
from the top, and its moment of inertia with respect to the axis ^^z 
17.08 inches*. Ans. 1 I5.s iiu-hes\ 

2. Compute the moment of ineilia of tlie section of Fig. 31, (f, 



54 



STKENGTH OF MATEEIALS 



with respect to tlie axis BB. Given that distance of the center 
of gravity of one angle from one side is 1.66 inches (see Fig. 31, J), 
and its moment of inertia with respect to hh 17.68 inches. 

Ans. 77,5 inches\ 
55. Table of Centers of Gravity and floments of Inertia. 
Column 2 in Table A below, gives the formula for moment of 
inertia with respect to the horizontal line through the center of 
gravity. The numbers in the third column are explained in Art. 
62; and those in the fourth, in Art. 80. 

TABLE A. 

Moments of Inertia, Section Moduli, and Radii of Gyration. 

In each case the axis is horizontal and passes through the center of gravity. 



Section. 








,,. 




> a» 




. .i_ 





-\>- 


' 






ft. 


T" 


. , 


-T— 


t^ 


^bi 




-J- 


-[tof 


_L 




Moment of 
Inertia. 



12 



a^ - a^^ 



12 



ba^ 
12 



ba^-bja,^ 
12 " 



0.019d4 



0.049 (d-* - d/) 



Section 
Modulus. 



a^ - aj 
6a 

ba2 
6 



ba'^-b^aj' 
6a 



0.098d3 



0.098- 



d4-d* 



Radius of 
Gyration. 



a 


1/ T2 


1 a2 + a,-^ 
\ 12 

a 


r'i2 


1 ba-^-b,a,^^ 



12(ba-b^aj) 
4 



l/d^+d^^' 



STRENGTH OF BEAMS. 

56. Kinds of Loads Considered. The loads that are applied 
to a horizontal beam are usually vertical, but sometimes forces are 
applied otherwise than at right angles to beams. Forces acting on 
beams at right angles are called transverse forces ; those applied 



STREKGTPI OF MATERIALS 



DO 



TABLE B. 

Shear Diagframs (b) and Moment Diagframs (c) lor Eight Different Cases (a). 
Also Values of Maximum Shear (V), Bending floment (M), and Deflection (d). 



' L 

A. 1 — 



&. c 



■W-uniform loek.d 



'inil!i|||i||||i]|||!!!|jl|||jj|][||j[i|| 



V=P, M=:P1, d=Pl3-f-3EI. 



a. X- 




Y=MP, M=i^Pl, d=Pl3H-48EI. 



^ 



- ftw 



i^ 




V=Pa^l, M=Pab^l. 



a- 



V=><;P, M=v/«P1, (1 = P1»--192EI. 






"-=-™iiniiiniiii]iil^^ 



V=W, M=3^W1, d=Wl3-f-8EI. 



2^ f= 



V7-ciniform loa-d 



h II^I^^T^^^^I^^^ _ 




V=J^ W, M=%W1, d=5Wl3^384EI. 



,P .P , 



V=P, M=Pa, d-Pa(3l2-4a2)-=-24EI. 



"W-uniform loo^ 






56 



STEEKGTH OF MATERIALS 



parallel to a beam are called Idngitiidinal forces ; and others are 
called inclined forces. For the present we deal only with beams 
subjected to transverse forces (loads and reactions ). 

57. Neutral Surface, Neutral Line, and Neutral Axis. When 
a beam is loaded it may be wholly convex up (concave down), as a 
cantilever; wholly convex down (concave up), as a simple beam 
on end supports; or partly convex up and partly convex down, as 
a simple beam with overhanging ends, a restrained beam, or a con- 





N/ 



Fig. 32. 

tinuous beam. Two vertical parallel lines drawn close together on 
the side of a beam before it is loaded will not be parallel after it 
is loaded and bent. If they are on a convex-down portion of a 
beam, they will be closer at the top and farther apart l)elow than 
when drawn (Fig. 32^/), and if they are on a convex-up portion, 
they will be closer below and farther apart above than when drawn 
(Fig. 32//). 

The '• fibres '' on the convex side of a beam are stretched and 
therefore under tension, while those on the concave side are short- 
ened and -therefore under compression. Obviously there must be 
some intermediate fibres which are neither stretched nor shortened, 
i, e., under neither tension nor compression. These make up 
a sheet of fibres and define a surface in the beam, which surface is 
called the neutral surface of the beam. The intersection of the 
neutral surface with either side of the beam is called the neutral 
line, and its intersection with any cross-section of the beam is 
called the neutral axis of that section. Thus, if aJj is a fibre that 
has been neither lengthened nor shortened with the bending; of the 
beam, then nn is a portion of the neutral line of the beam; and, 
if Fig. 32c be taken to represent a cross-section of the beam, JS^X 
is the neutral axis of the section. 

It can be proved that the neutral axis of any cross -section of 



STRENGTH OF MATERIALS 



57 



a loaded beam jpasses througJt the center of gravity of that section^ 
provided that all the forces applied to the beam are transverse, and 
that the tensile and compressive stresses at the cross-section are 
all within the elastic limit of the material of the beam. 

58. Kinds of Stress at a Cross=section of a Beam. It has 
already been explained in the ])receding article that there are ten- 
sile and compressive stresses in a beam, and tliat the tensions are 
on the convex side of the beam and the compressions on the con- 
cave (see Fig. 88). The forces T and are exerted upon the 
portion of the beam represented by the adjoining portion to the 



> 


or 




< 

















I- c 

^ — 

«- 
e- 

■♦ 

I^ T 

— ► 
1 — y 




1 


or 

























-^ T 

— > 

♦ 

h c 
* — 



e- 



FifT. 33. 



right (not shown). These, the student is reminded, are often called 
fibre stresses. 

Besides the fibre stresses there is, in creneral, a shearincr stress 
at every cross-section of a beam. This niay be proved as follows: 

Fig. 84 represents a simple beam on end supports which has 
actually been cut into two parts as shown. The two parts can 
maintain loads ^\•ll(Ml in a liorizontal position, if forces are applied 
at the cut ends equivalent to the forces tliat would act there if the 
beam were not cut. Evidently in the solid beam there are at the 
section a compression above and a tension bi'low, and such forces 
can be a])[)lied in the cut beam by means of a short block C and a 
chain or cord T, as shown. The block furnishes the compressive 
forces and the chain the tensile forces. At first sight it aj)pears as 
if the beam would stiind up under its load after the block and 
chain have been put into place. Excej)t in certain cases*, how- 
ever, it would not remain in a horizontal jmsition, as would the 



* WIkui tlu; oxtcrnal slusir for tlu^ sci-ticm is vawo. 



58 



STEEXGTII OF MATEKIALS 



solid beam. This shows that the forces exerted by the block and 
chain (horizontal compression and tension) are not equivalent to 
the actual stresses in the solid beam. What is needed is a vertical 
force at each cut end. 

Suppose that Ej is less than L^ -f-L^ + ^'^ight of A, i. e.^ that 
the external shear for the section is negative; then, if vertical pulls 
be applied at the cut ends, upward on A and downward on B, the 
beam will stand under its load and in a horizontal position, just as 
a solid beam. These pulls can be supplied, in the model of the 
beam, by means of a cord S tied to two brackets fastened on A and 



T/y/ 



?^^i 



B 



^^7^\ 



Fig. 34. 



W^ 



^S 



W 



"c 



Fig. 35. 



Ri 



B, as shown. In the solid beam the two parts act upon each 
other directly, and the vertical forces are shearing stresses, since 
they act in the plane of the surfaces to which they are ap])lied. * 

59. Relation Between the Stress at a Section.and the 1-oads 
and Reactions on Either Side of It. Let Fig. 35 represent the 
portion of a beam on the left of a section; and let E^ denote the 
left reaction; L^ and L, the loads; W the weight of the left part; 

C, T, andS the compression, tension, and shear respectively which 
the riaht part exerts upon the left. 

Since the part of the beam here represented is at rest, all the 
forces exerted upon it are balanced; and when a number of hori- 
zontal and vertical forces are balanced, then 

1. The algebraic sum of the horizontal forces equals zero. 

2. " " " " " vertical " " " 

3. " " " " " moments of all the forces with respect to 
any point equals zero. 

To satisfy condition 1, since the tension and compression are 

the only horizontal forces, the tension must eqiLal the compression . 

To satisfy condition 2, S (the internal sliear) must equal the 



STRENGTH OF MATERIALS 59 

algebraic sum of all the other vertical forces on the portion, that 
is, must ecjual the external shear for the section; also, S must act 
up or down according as the external shear is negative or ])Ositive. 
In other words, briefly expressed, the internal and external shears 
at a section are equal and ojyjjosite. 

To satisfy condition 3, the algebraic sum of the moments of 
the fibre stresses about the neutral axis must be equal to the sum 
of the moments of all the other forces acting on the portion about 
the same line, and the signs of those sums must be opposite. (The 
moment of the shear about the neutral axis is zero.) Xow, the 
sum of the moments of the loads and reactions is called the bend- 
ing moment at the section, and if we use the term resisting mo= 
ment to signify the sum of the moments of the fibre stresses (ten- 
sions and compressions ) about the neutral axis, then we may say 
briefly that the resisting and the bending moments at a section are 
equals and the tico moments are oi:>posite in sign. 

60. The Fibre Stress. As before stated, the fibre stress is 
not a uniform one, that is, it is not uniformly distributed over the 
section on which it acts. At any section, the compression is most 
"intense" (or the unit-compressive stress is greatest) on the con- 
cave side; the tension is most intense (or the unit-tensile stress is 
greatest) on the convex side; and the unit-compressive and unit- 
tensile stresses decrease toward the neutral axis, at which place the 
unit-fibre stress is zero. 

If the fibre stresses are within the elastic limit, then the two 
straight lines on the side of a beam referred to in Art. 57 will still 
be straight after the beam is bent; hence the elongations and short- 
enings of the fibres vary directly as their distance from the neutral 
axis. Since the stresses (if within the elastic limit) and deforma- 
tions in a given material are proportional, tJie anit-jihre stress 
varies as the distance fi'ont the neutral (fxis. 

Let Fig. "i^Ga represent a portion of a bent beam, 3()/> its cross- 
section, jui ihi) neutral lini^, and NN the niMitral axis. The way 
in which the unit-fibre stress on tlie section vai-ies can he ren 
resented graphically as follows: Lay olf cr, by some scale, to 
represent the unit-tibre stress in the top iibre, and join c and /f. 
extending the line to tliti lower side of the beam ; also make be' equal 
to Ar" and draw //<■'. Then the ari'ows repi'esi'iit the unit-tibre 
Btresties,for their lengths vary as their distances from the neutral axis. 



60 



STKENGTH OF MATEKIALS 



6i. Value of the Resisting Moment. If S denotes the unit - 
fibre stress in the fibre farthest from the neutral axis (the greatest 
unit-fibre stress on the cross-section), and c the distance from the 
neutral axis to the remotest fibre, Avhile S^, Sg, S3, etc., denote the 
unit-fibre stresses at points whose distances from the neutral axis 
are, respectively, y„ 2/3, 7/3, etc. (see Fig. 36^), then 

S : Sj :: r : y,\ or S^ = -^y,. 



Also, 



S. 



-- 2/2; ^3 =—2/35 «tc. 



Let «j, «2j ^'-z') ^tc, be the areas of the cross-sections of the fibres 




ac To 

Fig. 36. 

whose distances from the neutral axis are, respectively, y^^ y.,^ 2/3, 
etc. Then the stresses on those fibres are, respectively, 

Si a,, S2 a.,, S3 (f^, etc.; 
S S S 



or 



n -y/^2y jy/h^ etc. 



The arms of these forces or stresses with respect to the neutra-i axis 
are, respectively, t/j, ?/,, ?/,, etc.; hence their moments are 

s ' \ s ^^ ^^^ s 






a^l, etc., 



and the sum of the moments (that is, the resisting moment) is 

- —^1 yi H — r^^2 yl -f etc. =-r{<'i y] -^ ^>i y^ + etc.) 

Xow a^ y\ + ^f., y\ + etc. is the sum of the px'oducts obtained by 
multiplying each infinitesimal part of the area of the cross-section 
by the square of its distance from the neutral axis; hence, it is the 
moment of inertia of the cross-section with respect to the neutral 
axis. If this moment is denoted by I, then the value of the resist- 

^ . SI 
in or moment is — 






d 


, 


ei 


tJ 


i-> 


^ 





>-^ 




o 


l;^ 


o 


s 


< 


^ 


u 


*-l 






X 


r^^ 


o 


0) 


« 


<u 


o 


Pi 

a 


o 




n 


^ 




Q 


W) 


I-] 


•S 


M 


2 


^ 


s 


iJ 


K 


i-l 




< 


o 


X 


tj) 


i/i 


-r-l 


< 




b 


73 

+2 


o 


O 


H 


*^ 


O 




H 


<1 


to 




hJ 


n 


>— 1 

<: 


O 


H 


^j 


M 




w 






-» 




















;^ 




s 




P3 



STRENGTH OF MATERIALS, 

PART II. 



STRENGTH OF BEAnS===(Concluded). 

62. First Beam Formula. As sliowii in the ])receding 
article, the resisting and bending moments for an^' section of a 
beam are equal; hence 

? = ^^A (6) 

all the symbols referring to the same section. This is the most 
important formula relating to beams, and will be called the '• first 
beam formula." 

The ratio I -^- e is now quite generally called the section 
modulus. Observe that for a given beam it depends only on the 
dimensions of the cross-section, and not on the material or any- 
thing else. Since I is the product of four lengths (see article 51), 
I -^ <? is the product of three; and hence a section modulus can be 
expressed in units of volume. The cubic inch is practically always 
used; and in this connection it is written thus, inchesl See Table 
A, page 54, for values of the section moduli of a few simple sections. 

63. Applications of the First Beam Formula. riui-e are 
three principal a])plications of ecpiation (>, which will now l)e ex- 
plained and illustrated. 

64. Vii'st A])pli('((t'((>]\ . The dinu'usions of a beam :iiid its 
manner of loading and suj)[)ort ai\' gi\eu, and it is r('(juii\'d to 
compute the greatest unit-tensile and conipressiN c stri'sses in the 
beam. 

This problem can be soKcd 1»\ means of ('([uat ion <), written 
in (his form, 

S- -por jf. ^, (6') 

Unless otherwises stated, we assnmc tliat tbc iieams arc nnifoi'm 
in cross-section, as tiic\ nsnally arc; then the section motlulus 
(1 : r) is tlic sanu- for all sections, and S ( the nnit libi-c stress on 



62 STEEXGTH OF MATEEIALS 



the remotest fibre) varies iust as M varies, and is therefore o-reatest 
where M is a maximiirQ.* Hence, to compute the value of tlie 
greatest unit-fibre stress in a given case, substitute the valutas of 
the section modulus and the maximum hending moment in the 
preceding ec[uatioii „ and reduce. 

If the neutral axis is eq^ually distant from the highest and low- 
est fibres, then the greatest tensile and compressive unit-stresses 
are equal, and their value is S. If the neutral axis is unequally 
distant from the highest and lowest fibres, let c denote its distance 
fi'om the nearer of the two, and S' the unit-fibre stress there. 
Then, since the unit-stresses in a cross-section are proportional to 
the distances from the neutral axis, 

S' c' , c' 

TT- = — . or S' = - — S. 

b c ' c 

If the remotest fibre is on the convex side of the beam, S is tensile 
and S' compressive; if the remotest fibre is on the concave side, S 
is compressive and S' tensile. 

Ji^xamjyJes. 1. A beam 10 feet long is supported at its ends, 
and sustains a load of 4,000 pounds two feet from the left end 
•(Fig. 37. a^). If the beam is 4 X 1- inches in cross-section (the 
long side vertical as usual), compute the maximuin tensile and 
compressive unit-stresses. 

The section modulus of a rectangle whose base and altitude 
are h and a respectively (^se^j Table A. page 54), is J^-^'"; hence, 
for the beam under consideration, the moditlus is 

-- X 4 X 12' = 'M') inchest 

To compute the maximum bending moment, we have, first, to find 
the dangerous section. This section is where the shear changes 
sign (see article 45); hence, we have to construct the shear dia- 
DTam. or as much thereof as is needed to find where the change of 
sicrn occurs. Therefore Ave need the values of the reaction. 
Xecrlectincr the weio-ht of the beam, the moment etiuation with 
origin at C (Fig. 37. a\ is 

R, X 10 - 4.000 X 8 = 0, or E, = 3,200 pounds 

^ Note. Because S is greatest in the section where M is maximum, this 
section is usuallv called the '• dangerous section'" of the beam. 



STRENGTH OF MATERIALS 



63 



Then, constructing the shear diagram, we see (Fig. 37, h) that the 
change of sign of the shear (also the dangerous section) is at the 
load. The value of the bendingr moment there is 

3,200 X 2 == 0,4:00 foot-pounds, -- 

or G,400 X 12 = 76,800 inch-pounds. 

Su])stitutiiig in ecpiation 6', we find that 

8 = — ^. — - = 800 pounds per square inch. 
4ooolbs. 




■io(a) 



c' (b) 



c" (c) 



FiL^ 37. 



2, It is desired to take into account the weight of the heani 
in the ])receding example, supposing the beam to he wooiUmi. 
The volume of the beam is 

-— —- X 10 r=- 3.\ cubic feet; 

and supposing the timber to weigli 45 pounds per cubic foot, the 
beam weighs 150 pounds (insignificant compartMl to the l(^adV 
The left reaction, therefore, is 



;.j;2()()-f(_X 150) -=3,275; 



64 STKENGTH OF MATEEIALS 

and the shear diagram looks like Fig. 37, <?, the shear changing 
sign at the load as before. The weight of the beam to the left of 
the dangerous section is BO pounds; hence the maximum bending 
moment equals 

3,275 X 2 - 30 X 1 = 0,520 foot-pounds, 

or 0,520 X 12 = 78,240 inch-pounds. 

Substituting in equation G', we find that 

o ^8,240 ^^^ ^ . ^ 

b = — ^T7^ — := ol5 pounds per square inch. 

The weight of the beam therefore increases the unit-stress pro- 
duced by the load at the dangerous section by 15 pounds per 
square inch. 

3. A T-bar (see Fig. 38) 8 feet long and supported at each 



"^(jT" end, bears a uniform load of 1,200 



N -^- 



h: 



d -j^ pounds. The moment of inertia of its 

I cross -section with respect to the neu- 
"cO tral axis being 2.42 inches*, compute 
oi the maximum tensile and compressive 
^ ■ unit- stresses in the beam 



^' ' Evidently the dangerous section 

is in the middle, and the value of the maximum bendint>- moment 
(see Table B, page 55, Part I) is J Wl, W and I denoting the load 
and length respectively. Here 

« 

-^ ^Yl = -g- X 1,200 X 8 = 1,200 foot-pounds, 

or 1,200 X 12 = 14,400 inch-pounds. 

The section modulus equals 2.42 -^ 2.28 == 1.06; hence 

b = ^ ..,. = lo,050 pounds per square inch. 

This is the unit-fibre stress on the lowest fibre at the middle sec- 
tion, and hence is tensile. On the highest fibre at the middle 
section the unit- stress is compressive, and equals (see page 62) : 

^ == — ^ = ij-;jn X 13,585 = 4,290 ])ounds ])er square inch. 



STRENGTH OF MATERIALS 05 

EXAMPLES FOR PRACTICE. 

1. A beam 12 feet long and G X 12 inches in cross -section 
rests on end supports, and sustains a load of 8,000 pounds in the 
middle. Compute the greatest tensile and compressive unit- 
stresses in the beam, neglecting the weight of the beam. 

Ans. 750 pounds per square inch. 

2. Solve tlie preceding example taking into account the 
weight of the beam, 800 pounds 

Ans. 787.5 pounds per square inch. 

3. Suppose that a built-in cantilever projects 5 feet from the 
wall and sustains an end load of 250 pounds. The cross -section of 
the cantilever being represented in Fig. 38, compute the greatest 
tensile and compressive unit-stresses, and tell at what places they 
occur. (Neglect the weight.) 

j Tensile, 4,469 pounds per square inch. 

^^' \ Compressive, 14,150 " " '• " 

4. Compute the greatest tensile and compressive unit-stresses 
in the beam of Fitr. 18, a, due to the loads and the weitrht of beam 
(400 pounds). (A moment diagram is represented in Fig. 18, I; 
for description see example 2, Art. 44, p. 31).) The section of 
the beam is a rectangle 8 X 12 inches. 

Ans. 580 pounds ])er sijuare inch. 

5. Compute the greatest tensile and compressive unit-stresses 
in the cantilever beam* of Fig. lU, a, it being a steel I-beam whose 
section modulus is 20.4 inches'^ (A bending moment diao-ram for 
it is represented in Fig. 10, h; for description, see ]*x. 3, Art. 44.) 

Ans. 11,470 pounds per square inch. 

G. Compute the greatest tensile and com])ressive unit-stresses 
in the beam of Fitr. 10 neo-lectino; its weight, the cross-sections 
being rectangular G X 12 inches. (See example for practice 1, 
Art. 43.) 

Ans. GOO pounds por sipiarc inch. 

65. Second Aj>j>l iadloii . The dimensions and tlu^ work- 
ing strengths of a beam are given, and it is recpiircd to dcltM-niine 
its safe load (the manner of application being given). 

This problem can lu^ solved by means of iMjnation G writtiMi 
in this forni. 



G 



66 STEENGTir OF MATEEIALS 

We substitute for S the given working strength for the ma- 
terial of the beam, and for I and c their values as computed from 
the given dimensions of the cross-section; then reduce, thus 
obtaining the value of the safe resisting moment of the beam, 
which equals the greatest safe bending moment that the beam can 
stand. We next compute the value of the maximum bending 
moment in terms of the unknown load; equate this to the value 
of the resisting moment previously found; and solve fo^ the 
unknown load. 

In cast iron, the tensile and compressive strengths are very 
diiferent; and the smaller (the tensile) should always be used if 
the neutral surface of the beam is midway between the top and 
bottom of the beam; but if it is unequally distant from the top 
and bottom, proceed as in example 4, following, 

ExamjDles. 1. A wooden beam 12 feet long and 6 X 12 
inches in cross -section rests on end supports. If its working 
strength is 800 pounds per square inch, how large a load uniformly 
distributed can it sustain ? 

The section modulus is -J^«^, h and a denoting the base and 
altitude of the section (see Table A, page 51); and here 

i Ur =-7rX X 12' =-- HI inchesl 
o b 

Hence S — = 800 X 114 = 115,200 inch-pounds. 

For a beam on end supports and sustaining a uniform load, the 
maximum bending moment equals |^WZ (see Table E, page 55), 
W denoting the sum of the load and weight of beam, and I the 
length. If W is expressed in pounds, then 

11 1 

W/ = -^ W X 12 foot-pounds = - W X 144 inch-pounds. 

Hence, equating tb<i two values of maximum bending moment 
and the safe resisting moment, we get 

1 

8 



^Y X 144 = 115,200; 



_ 115,200 X 8 , ,^^ 
or, W = '-j^ = 6,400 pounds. 



STRElS^GTir OF MATERIALS 



67 



The safe load for the beam is 6,400 pounds iiiiiius the weight of 
the beam. 

2. A steel I-beam whose section modulus is 20.4: inches' 
rests on end supports 15 feet apart. Xeglecting the weight of the 
beam, how large a load may be placed upon it 5 feet from one end, 
if the working strength is 16,000 pounds per square inc^h? 

The safe resistin^jf moment is 



SI 

G 



16,000 X 20.4 = 326,400 inch-pounds; 



hence the bendino; moment must not exceed that value. The 
dangerous section is under the load; and if P denotes the unknown 
value of the load in pounds, the maximum moment (see Table B, 
page 55, Part I) equals § P X 5 foot-pounds, or § P X 60 inch- 
pounds. Equating values of bending and resisting moments, 
we get 



-^ P X 60 = 326,400; 



or. 



P 



326,400 X 3 



8,1()() pounds. 



3. In the preceding example, it is required to take into 
account the weight of the beam. 375 pounds. 



5'- 



P 



lO- 



R. 



'W=375l"bS. 



Fig. 39. 



As we do not know tlu? value of tlu^ sni'v l(nul, we cannot con- 
struct the shear diaoram and thus determine wIutc the (lano:erous 
section is. But in cases like this, wluM*e the distributed load (the 
weight) is small compared with the concentrated load, the dan- 
gerous section is j)ractically always wnere it is under the concen- 
trated load alone; in this case, at the load. The reactions due to 
the weight e(jual I X 375 = 1S7.5; aiul the reactions due to tlu> 
load e(pud .\ P and | P, P denoting the value of the load. The 
larger reaction 11, (Fig. 30) heiu'e ecpuils :^ P | 1S7.5. Since 



68 STRENGTH OF MATEEIALS 

the weight of the beam per foot is 375 -f- 15 = 25 pounds, the 
maximum bending moment (at the load) equals 



( ^ P -L 187.5) 5 - (25 X 5) 2i = 

-T^ P -I- 937.5 - 312.5 = -rr P -f 625. 

This is in foot-pounds if P is in pounds. 

The safe resisting moment is the same as in the preceding 
illustration, 326,100 inch-pounds; hence 

(^ P + 625) 12 = 326,400. 

Solvincr for P, we have 

10 -p , ... 326,400 

10 P H- 625 X 3 - '^^^'"^^^ ^ ^ = 81,600; 

10 P = 79,725; 

or, P = 7,972.5 pounds. 

It remains to test our assumption that tlie dangerous section 
is at the load. This can be done by computing K^ (with P = 
7,972.5), constructincr the shear diagrram, and notinp; where the 
shear chancres sig-n. It will be found that the shear chancres sicrn 
at the load, thus verifying the assumption. 

4. A cast-iron built-in cantilever beam projects 8 feet from 
the wall. Its cross-section is represented in Fig. 40, and the 

moment of inertia with respect to 
the neutral axis is 50 inches*; the 

]vj (__| 5f-N working strengths in tension and 

compression are 2,000 and 9,000 
pounds per square inch respect- 
ively. Compute the safe uniform 
Fiff 40 J r 

load which the beam can sustam, 

necrlectinor the weight of the beam. 

The beam being convex up, the upper fibres are in tension 
and the lower in compression. The resisting moment (SI -^ c), 
as determined by the compressive strength, is 









1 -^ 








■?^ in 

— |-N 
lo 




1 



STRENGTH OF MATERIALS 69 

9,000 X 50 ,,,„„„.. , , 

— — j-p. =: 100,000 inch-pounds; 

and the resisting moment, as determined by the tensile strength, is 
2,000 X 50 



2.5 



40.000 inch-pounds. 



Hence the safe, resisting moment is the lesser of these two, or 
40,000 inch-pounds. The dangerous section is at the wall (see 
Table B, page 55), and the value of the maximum bending 
moment is ^ W/, W denoting the load and I the length. If AV is 
in* pounds, then 

M = 1 W X 8 foot-pounds = ^ ^Y x 9G inch-pounds. 

Equating bending and resisting moments, we have 

4-W X 96 = 40,000; 

40,000 X 2 
or, W = ^— = 8dd pounds. 

EXAMPLES FOR PRACTICE. 

1. An 8 X 8-inch timber projects 8 feet from a wall. If its 
working strength is 1,000 pounds per square inch, how large an 
end load can it safely sustain ? 

Ans. 800 pounds. 

2. A beam 12 feet long and 8 X 10 inches in cross-section, 
on end supports, sustains two loads P, each 3 feet from its ends 
respectively. The working strength being 1.000 pounds per square 
incii, compute P (see Table B, })age 55). 

Ans. 0,480 ])ounds. 

3. An I-beam weighing 25 pounds ])er foot rests on end 
supports 20 feet apart. Its section modulus is 20.4 inches^ and 
its working strength 1(),000 pounds ])er squaiv incli. Compute 
the safe uniform load which it can sustain. 

Ans. 10,SS0 pounds- 

66. Tli'n'd A]>j)li<'<ffi()n . Tlu* k)ads, manner of support, 

and working strength of beam are givcMi, and il is rtMpiirtHl (o dc- 

termine the size of cross-section necessary to sustain the load 

safely, that is, to "design the bivim/' 



70 STRENGTH OF MATERIALS 

To solve this problem, we use the first beam formula (etjuation 
G), writteu iu this form, 

J^ _M. (6'") 

c S 

We first determine the maximum bending moment, and then sub- 
stitute its value for M, and the working Btrenorth for S. Then we 
have the value of the section modulus (I ^- r) of the required 
beam. Many cross-sections can be designed, all having a given 
section modulus. Which one is to be selected as most suitable will 
depend on the circumstances attending the use of the beam and 
on considerations of economy. 

Examples. 1. A timber beam is to be used for sustaining 
a uniform load of 1,500 pounds, the distance between the supports 
being 20 feet. If the working strength of the timber is 1,000 pounds 
per square inch, what is the necessary size of cross-section ? 

The dano-erous section is at the middle of the beam; and the 
maximum bending moment (see Table B, page 55) is 

—-^l = -rr X 1,500 X 20 = 3,750 foot-pounds, 
or 3,750 X 12 = 45,000 inch-pounds. 

TT ^ 45,000 ,. . , 3 

Hence ■ — = -. ,.,.,. = 4u inches . 

c l,UUi) 

ISTow the section modulus of a rectangle is Iha^ (see Table A, 

page 54, Fart I); therefore, -jZ'^r = 45, or da- = 270. 

Any wooden beam (safe strength 1,000 pounds per square 
inch) whose breadth times its depth square equals or exceeds 270, 
is strong enough to sustain the load specified, 1,500 pounds. 

To determine a size, we may choose any value for h or a, and' 
solve the last equation for the unknown dimension. It is best, 
however, to select a value of the breadth, as 1, 2, 3, or 4 inches, 
and solve for a. Thus, if we try 1 = 1 inch, we have 

a'^ = 270, or a = 16.43 inches. 
This would mean a board 1 X 18 inches, which, if used, would 
have to be supported sidewise so as to prevent it from tipping or 
" buckling." Ordinarilv, this would not be a good size. 

Next try 1 = 2 inches; we have 
2 X «' = 270; or a = i/270 ^ 2 = 11.62 inches. 
This would require a plank 2 X 12, a better proportion than the 
first. Trying I = 3 inches, we have 



STKEN^GTIl OF MATEEIALS 



3 X ((^^ = 270; or a = V 210 -^ ;j ^ ^'.4:'.) inches. 

This would require a plank 3 X 10 inches; cind a choice between 
a 2 X 12 and a 3 X 10 plank would be governed bv circumstances 
in the case of an actual construction. 

It will be noticed that we have neo-lected the weio-lit of the 
beam. Since the dimensions of wooden beams are not fractional, 
and we have to select a commercial size next laro-er than the one 
computed (12 inches instead of 11.62 inches, for example), the 
additional depth is usually sufficient to provide strength for the 
weight of the beam. If there is any doubt in the matter, we can 
settle it by computing the maximum bending moment including 
the weight of the beam, and then computing the greatest unit-fibre 
stress due to load and weityht. If this is less than the safe streno-th, 
the section is large enough; if greater, the section is too small. 

Thus, let us determine whether the 2 X 12-inch ])lank is 
Rtroncr enouo-h to sustain the load and its own weicrht. The plank 
will weigh about 120 pounds, making a total load of 

1,500 + 120 = 1,620 pounds. 

Hence the maximum bendinor moment is 

o 

-Ly^i = —1,620 X 20 X 12 = 48,600 inch-pounds. ' 

Since — = -77- h(r -= —-X 2 X 12' = 48, and 8 = -j^. , " 

48,600 ^ ^ ^ 
o = — J— ^^1,013 pounds per square inch. 

Strictly, therefore, the 2 X 12-inch plank is nDt large enough; but 
as the greatest unit-stress in it would be only 13 ])ounds per sciuare 
inch too large, its use would be permissible. 

2. What size of steel I-beam is needed to sustain safely the 
loading of Fig. if the safe strength of the steel is 16,000 pounds 
per s(]uare inch ? 

.The maximum bending moment due to the loads was found 
in examj)le 1, Art. 43, to be S,SOO ro()t-ju)un(ls. or S,S()() \ 1:2 _^ 
105,600 incli-pounds. 

1 105,600 

That is, an l-iu'ani is needed whose section modulus is a little 
larger than ().(>, lo provide strength for its own weioht. 



72 



STKEXGTH OF MATERIALS 



To select a size, we need a descriptive table of I-beams, such 

as is published in handbooks on structural steel. 

Below is an abridged copy of such a table. (The last two columns con- 
tain information for use later.) The figure illustrates a cross-section of an 
I-beam, and shows the axes referred to in the table. 

It will be noticed that two sizes are given for each depth; 
these are the lightest and heaviest of each size that are made, but 
intermediate sizes can be secured. In column 5 we find 7.3 as the 
next larger section modulus tlian the one required {Q-Q)', and this 
corresponds to a 12^-pound 6-inch I-beam, which is probably the 
proper size. To ascertain whether the excess (7.3-6.6 = 0.70) 
in the section modulus is sufficient to provide for the weicrht of the 
beam, we might proceed as in example 1. In this case, however, 
the excess is quite large, and the beam selected is doubtless safe. 

TABLE C. 
Properties of Standard I^Beams 

1' 



1 



Section of beam, showing axes 1-1 and 2-2. 



1 


2 


3 


i 


5 


6 


Depth of 


Weight 


Area of cross- 


Moment of 


Section 


Moment of 


Beam. 


per foot. 


section, in 


inertia. 


modulus. 


inertia, 


in inches. 


in pounds. 


square inches. 


axis 1—1. 


axis 1—1. 


axis 2—2. 


3 


5.50 


1.&3 


2.5 


1.7 


0.46 


3 


7.50 


2.21 


2.9 


1.9 


.60 


4 


7.50 


2.21 


6.0 


3.0 


.77 


4 


10.50 


3.09 


7.1 


3.6 


1.01 


5 


9.75 


2.87 


12.1 


4.8 


1.23 


5 


14.75 


4.34 


15.1 


6.1 


1.70 


6 


12.25 


3.61 


21.8 


7.3 


1.85 


6 


17.25 


5.07 


26.2 


8.7 


2.36 


7 


15.00 


4.42 


36.2 


10.4 


2.67 


7 


20.00 


5.88 


42.2 


12.1 


3.24 


8 


18.00 


5.33 


56.9 


14.2 


3.78 


8 


25.25 


7.43 


68.0 


17.0 


4.71 


9 


21.00 


6.31 


84.9 


18.9 


5.16 


9 


35.00 


10.29 


111.8 


24.8 


7.31 


10 


25.00 


7.37 


122.1 


2i.4 


6.89 


10 


40.00 


11.76 


158.7 


31.7 


9.50 


12 


31.50 


9.26 


215.8 


36.0 


9.50 


12 


40.00 


11.76 


245.9 


41.0 


10.95 


15 


42.00 


12.48 


441.8 


58.9 


14.62 


15 


60.00 


17.65 


538.6 


71.8 


18.17 


18 


• 55.00 


15.93 


795.6 


88.4 


21.19 


18 


70.00 


20.59 


921.2 


102.4 


24.62 


20 


65.00 


19.08 


1,169.5 


117.0 


27.86 


20 


75.00 


22.06 


1.268.8 


126.9 


30.25 


24 


80.0C» 


23.32 


2.087.2 


173.9 


42.86 


24 


100.00 


29.41 


2,379.6 


198.3 


48.55 



STEENGTH OF MATERIALS 73 

EXAMPLES FOR PRACTICE. 

1. Determine the size of a wooden beam which can safely 
sustain a middle load of 2,000 pounds," if the beam rests on end 
supports 16 feet apart, and its working strength is 1,000 pounds 
per square inch. Assume width inches. 

Ans. 6 X 10 inches. 

2. What sized steel I-beam is needed to sustain safely a 
uniform load of 200,000 pounds, if it rests on end supports 10 
feet apart, and its working strength is 16,000 pounds per square 
inch? 

Ans. 100-pound 24:.inch. 

3. What sized steel I-beam is needed to sustain safely the 
loading of Fig. 10, if its working strength is 16,000 pounds per 
square inch ? 

Ans. ll:.75-pound 5 -inch. 

67. Laws of Strength of Beams. The strength of a beam is 
measured by the bending moment that it can safely withstand; or, 
since bending and resisting moments are equal, by its safe resist- 
ing moment (SI -^ a). Hence the safe strength of a beam varies 
(1) directly as the working fibre strength of its material, and (2) 
directly as the section modulus of its cross-section. For beams 
rectangular in cross-section (as wooden beams), the section modu- 
lus is ^ha\ 1> and a denoting; the breadth and altitude of the 
rectangle. Hence the strength of such beams varies also directly 
as the breadth, and as the square of the depth. Thus, doubling 
the breadth of the section for a rectangular beam doubles the 
strength, but doubling the depth quadruples the strength. 

The safe load that a beam can sustain varies directly as its 
resisting moment, and depends on the way in which the load is 
distributed and liow tlie beam is suj)])orte(l. Thus, in the first 
four and last two cases of the ta])le on ]>age 55, 

M = IV, hence 
M = i W/, - 
M - j IV, - 
M == J AV7, - 
M = J IV, '^ 
M = tV W/, -' 



V 


SI 


/,', 


\\ 


2SI 


/r. 


V 


4SI 


/<', 


AV - 


SSI 


/,', 


r -= 


SSI 


/<•, 


W=: 


12S1 - 


- /(..', 



74 



STEE^'GTII OF MATERIALS 



Therefore the safe load in all cast-s varies inversely with the 
lenoth: and for the different cases the safe loads are as 1. 2. 4, 8, 
&. and 12 respectively. 

E-camplf. What is the ratio of the strengths of a plank '2 X 
lij ijjchtr? when placed edgewise and when placed flatwise on its 
supports \ 

When placed edgewise, the section modulus of the plank is 
\ X 2 X 1(^ = 33 J. and when placed flatwise it is J X 1(» X 2- = 
6|; hence its strengths in the two positions are as 33| to 6| 



resT)ectively, or as 5 to 1. 



EXA.MPLE FOR PRACTICE. 

What is the ratio of the safe loads for two beams of wood, 
one beincr 10 feet lono-, 3 X 12 inches in section, and havingr its load 
in the middle; and the other 8 feet long and 2x8 inches in section, 
with its load uniformly distributed. 

Ans. As 135 to 100. 

d^. .Modulus of Rupture. If a beam is loaded to destruction, 
and the value of the bending moment for the rupture stao-e is 
computed and substituted for M in the formula SI ^- '. = M. then 
the value of S computed from the equation is the modulus of 
rupture for the material of the beam. Many experiments have 
been performed to ascertain the moduli of rupture for different 
materials and for different gi-ades of the same material. The fol- 
owincr are fair values, all in jxjuuds j»er square inch; 

TABLE D. 

Moduli of Rupture. 



Timhtr: 

Spruce 

Hemlock 

White pine 

Long-leaf pine.. . . 
Short-leaf pine . . . 
Douglas spruce.. . 

White oak 

Red oak 


4.C00— 7.000, average oSXKJ 

3.510 7.000, " 4.500 

5.500 10.500, « S.OOO 

10.000 16.CO0, " 12.500 

. 8.000 14.000, « 10.000 

' 4.000 12.000; '• 8.C3C30 

7.500 18!500; " 13.0CX) 

9.rr!0 i5.cx:30. " ufjco 


Stone : 

Sandstone 

Limestone 

Granite 


4<A»— 1.2UC», 
4CS0 1.000. 
800 i.4O0. 


Ca^i iron : 


One and one-half to two and 
t»ne-quarter times its ulti- 
mate tensile strength. 


Hard steel : 


Varies from ir«j.<:»;ii:» to loflCW 



STEENGTII OF MATEKJALS 75 

Wrouglit iron and structural steels have no modulus of rup- 
ture, as specimens of those materials will " bend double," but not 
break. The modulus of rupture of a material is used principally 
as a basis for determining its working strength. The factor of 
safety <f a loaded heam is corajjuted by dividing the modulus 
of rupture of its onaterial hy iJte greatest unit-Hhre stress itt 
the heam. 

69. The Resisting fhear. The shearing stress'on a cross- 
section of a loaded beam is not a uniform stress; that is, it is not 
uniformly distributed over the section. In fact the intensity or 
unit-stress is actually zero on the highest and lowest fibres of a 
cross-section, and is greatest, in such beams as are used in prac- 
tice, on fibres at the neutral axis. In the followino; article we 
explain how to find the maximum value in two cases — cases which 
are practically important. 

70. Second Beam Formula. Let Sg denote the average 
value of the unit-shearing stress on a cross-section of a loaded 
beam, and A the area of the cross-section. Then the value of the 
whole shearing stress on the section is : 

Resistino; shear = 8^ A. 

Since the resisting shear and the external shear at any section of a 
beam are equal (see Art. 59), 

S, A = V. (7) 

This is called the "second beam formula " It is used to investi- 
gate and to design for shear in beams. 

In beams uniform in cross-section, A is constant, and S^. ie 
greatest in the section for which V is greatest. Hence the great- 
est unit-shearing stress in a loaded beam is at the ni'utral axis of 
the section at which the external shear is a maximum. ThiMv is 
a formula for computing this nuiximum value in any case, but it 
is not simj)le, and we give a simpler metliod foi" computing the 
value in tlie two practically important cases: 

1. In vv()()(l(>n Ijcanis (r(H'i;inji:ular or S(|iicn(' in cross scctio?!"). tlir 
pr(Nit('st unit-slieariri'i;' stress in a section is HO jxm* (.-(mU 1ar<;rr than lht> avcrai^t^ 
valine S,. 

2. \\\ I beams, and in others with a thin verlii-al \\v\\ the •greatest 
unit sheariiij^ stress in a sei-tion |)ractically (>(|uals S^, as ^ixcn h\ e(|uation 7, 
if tho area of the web is sut)stituLed tor A. 



76 STKENGTH OF MATERIALS 



Examjyles. 1. What is the greatest value of the unit- 
shearing stress in a wooden beam 12 feet long and 6x12 inches in 
cross-section when resting on end supports and sustaining a uni- 
form load of 6,400 pounds ? (This is the safe load as determined 
by working fibre stress; see example 1, Art. 65.) 

The maximum external shear equals one-half the load (see 
Table B, page 55), and comes on the sections near the supports. 

Since A = 6 X 12 = 72 square inches; 

^s ^^ ^ryiy = ^^ pouuds per square inch, 
and the greatest unit-shearing stress equals 

-^ Sg = -^ 44 = 66 pounds per square inch. 

Apparently this is very insignificant; but it is not negligible, as 
is explained in the next article. 

2. A steel I-beam resting on end supports 15 feet apart 
sustains a load of 8,000 pounds 5 feet from one end. The weight 
of the beam is 375 pounds, and the area of its web section is 3.2 
square inches. (This is the beam and load described in examples 
2 and 3, Art. 65.) What is the greatest unit-shearing stress ? 

The maximum external shear occurs near the support where 
the reaction is the greater, and its value equals that reaction. 
Calling that reaction R, and taking moments about the other end 
of the beam, we have 

E X 15 - 375 X 7^ - 8,000 X 10 = 0; 

therefore 15 E = 80,000 + 2,812.5 = 82,812.5; 

or, R = 5,520.8 pounds. 

Hence Sg = ' i> o = 1,725 pounds per square inch. 

EXAMPLES FOR PRACTICE. 

1. A wooden beam 10 feet long and 2 X 10 inches in cross- 
section sustains a middle load of 1,000 pounds. Neglecting the 
weight of the beam, compute the value of the greatest unit-shearing 

stress. 

Ans. 37.5 pounds per square inch. 



STKENGTH OF MATERIALS 77 



2. Solve the preceding example taking into account the 
weight of the beam, 60 pounds. 

Ans. 40 pounds per square inch. 

8. A wooden beam 12 feet long and 4 X 12 inches in cross- 
section sustains a load of 8,000 pounds 4 feet from one end. 
IN^eglecting the weight of the beam, compute the value of the 
greatest shearing unit-stress. 

Ans. 62.5 pounds per square inch. 

71. Horizontal Shear. It can be proved that there is a 
shearing stress on every horizontal section of a loaded beam. An 
experimental explanation will have to suffice here. Imagine a 
pile of six boards of equal length supported so that they do not 
bend. If the intermediate supports are removed, they will bend 
and their ends will not be flush but somewhat as represented in 
Fig. 41. This indicates that the boards slid over each other during 
the bending, and hence there was a rubbing and a frictional re- 
sistance exerted by the boards upon each other. !Now, when a 
solid beam is being bent, there is an exactly similar tendency for 
the horizontal layers to slide over each other; and, instead of a 
frictional resistance, there exists shearing stress on all horizontal 
sections of the beam. 

In the pile of boards the amount of slipping is different at 
different places between any two boards, being greatest near the 
supports and zero midway between them. Also, in any cross- 
section the slippage is least between the upper two and lower two 
boards, and is greatest between the middle two. These facts indi- 
cate that the shearing unit-stress on horizontal sections of a solid 
beam is greatest in the neutral surface at the supports. 

It can be proved that at any place in a beam the shearing 
unit-stresses on a horizontal and on a vertical section are equal. 




I^'is. 41. Fipf. 42. 

It follows that the hoi'izontal shearinir unit-stress is iireatest at the 
neutral axis of tlie section for wliicli the external shear (V) is a 
maximum. Wood l)eino; very weak in slu'ar aloin'* the orain, 
timber beams sometimes fail under shear, the "rupture" being 



78 STEEJSTGTII OF MATERIALS 



two horizontal cracks along the neutral surface softiewhat as rep- 
resented in Fig. 42. It is therefore necessary, when dealing with 
timber beams, to give due attention to their strength as determined 
by the working strength of the material in shear along the grain. 

Example. A wooden beam 3 X 10 inches in cross-section 
rests on end supports and sustains a uniform load of 4,000 pounds 
Compute the greatest horizontal unit -stress in the beam. 

The maximum shear equals one-half the load (see Table B, 
page 55), or 2,000 pounds. Hence, by equation 7, since A = 
3 X 10 = 30 square inches, 

^, . 2,000 , /2 

bg = — ^Y\~ = Ob-TT- pounds per square inch. 

This is the averao-e shearincr uni -stress on the cross-sections near 
the supports; and the greatest value equals 

3 2 

— X 66— ^ 100 pounds per square inch. 

According to the foregoing, this is al&o the value of the 
greatest horizontal shearing unit-stress. (If of white pine, for 
example, the beam would not be regarded as safe, since the ulti- 
mate shearing strength alono- the grain of selected pine is only 
about 400 pounds per square inch.) 

72. Design of Timber Beams. In any case we may pro- 
ceed as follows: — [X) Determine the dimensions of the cross- 
section of the beam from a consideration of the fibre stresses as 
explained in Art. ^^. (2) With dimensions thus determined, com- 
pute the value of the greatest shearing unit-stress from the formula. 

Greatest shearincr unit-stress = -r ^" ^- <^f^h^ 

where V denotes the maximum external shear in the beam, and 
h and a the breadth and depth of the cross-section. 

If the value of the greatest shearing unit-stress so computed 
does not exceed the working streno-th in shear alona the grain, 
then the dimensions are large enough; but if it exceeds that value, 
then a or ^, or both, should be increased until f V -^ ah is less 
than the working strength. Because timber beams are very often 
''season checked" (cracked) along the neutral surface, it is advis- 



STKENGTII OF MATERIALS 79 



able to take the working strength of wocclen Ijeaiiis, in shear along 
the grain, quite low. One-twentieth of the working fibre strength 
has been Tecommended'^ for all pine beams. 

If the working strength in shear is taken equal to one- 
twentieth the working fibre strength, then it can be shown that, 

1. For a beam on end supports loaded in the middle, the safe load de- 
pends on the shearinj^ or fibre strength according as the ratio of length to 
depth {I -^ a) is less or greater than 10. 

2. For a beam on end supports uniformly loaded, the safe load depends 
on the shearing or fibre strength accordirg as Z -^ a is less or greater than 20. 

Examjjlt^'f. l-* It is required to design a timber beam to sus- 
tain loads as represented in Fig. 11, the working fibre strength 
being 550 pounds and the working shearing strength 50 pounds 
per square inch. 

The maximum bending moment (see example for practice 3, 
Art. 43; and example for practice 2, Art. 44) equals practically 
7,000 foot-pounds or, 7,000 X 12 = 84,000 inch-pounds. 
Hence, according to equation 6'", 

^ ^*-«0« = 152.7 inches3 



c 550 



Since for a rectane^le 

c b 

— hte = 152.7, or ba'' = 916.2. 
- u 

Now, if we let h = 4, then a^ =z 229; 

or, a = 15.1 (practically 16) inches. 

If, again, we let b = 6, then (("^ — 152.7; 

or a = 12.4 (practically 13) inches. 

Either of these sizes will answer so far as fibre stress is concerned, 
but tliere is more '•timber'' in tiie second. 

Tbc maxiniuiH external shear in tlie beam cipnils 1.556 
])Ounds, neglecting the weight of the beam (^see examj)Ie ',k \v\. 
37; and exnmplc 2, Art. 3<S). Therefore, for a 4 \ l()-in(.li beam, 

* See " Materials of Construction."— Johnson. Pago ^y^). 



80 STREXGTII OF MATEEIALS 

3 ^ 1.55G 
Greatest snearinor unit -stress =-^ X - — -— ^ 
^ :4 i X lo 

= 36.5 pounds per square inch; 
and for a 6 X 14- inch beam, it equals 

3 1.556 

~9~ ^ »^ — ~T1 ^^ pouncis per square inch. 

Since these values are less than the workino; streno-th in shear, 
either size of beam is safe as regards shear. 

If it is desired to allow for weight of beam, one of the sizes 
should be selected. First, its weight should be computed, then 
the new reactions, and then the unit-fibre stress may be com- 
puted as in Art. 64:, and the greatest shearing unit-stress as in the 
forecroinu. If these values are within the workincr values, then 
the size is large enouorh to sustain safely the load and the weio-ht 
of the beam. 

2. ^hat is the safe load for a white pine beam 9 feet long 
and 2 X 12 inches in cross-section, if the beam rests on end supports 
and the load is at the middle of the beam, the working fibre 
strencrth being l.UUU pounds and the shearing strength 5U pounds 
per square inch. 

The ratio of the length to the depth is less than 10; hence 
the safe load depends on the shearing strength of the material 
Callincr the load P. the maximum exteruc^d shear (see Table B, 
page 55) equals ^ P, and the formula for greatest shearing unit 
stress becomes 

3 .V P 
50 = ~o~ X q' -j 9 ; or P = l.OUU pounds. 

EXAMPLES FOR PRACTICE. 

1. "What size of wooden beam can safely sustain loads as in 
Ficr. 12. with shearing and fibre working strength equal to 50 and 
1,000 pounds per square inch respectively ? 

An 5. fj ■ 12 inches 

2. What is the safe load for a wooden beam 1 X 11 inches. 
and IS feet long, if the beam rests on end supports and the load 
is uniformly distributed, with working strengths as in example 1 ? 

Ans. 3,730 pounds 



STRENGTH OF MATERIALS 81 

73. Kinds of Loads and Beams. "We shall now discuss the 
strength of beams under longitudinal forces (acting parallel to 
the beam) and transverse loads. The longitudinal forces are 
supposed to be applied at the ends of the beams and along the axis* 
of the beam in each case. We consider only beams resting on 
end supports. 

The transverse forces produce bending or flexure, and the 
longitudinal or end forces, if pulls, produce tension in the beam; 
if pushes, they produce compression. Hence the cases to be con- 
sidered may be called " Combined Flexure and Tension " and 
" Combined Flexure and Compression." 

74. Flexure and Tension. Let Fig. 43, a^ represent a beam 
subjected to the transverse loads L,, Lg and L3, and to two equal 
end pulls P and P. The reactions Rj and Rg are due to the trans- 
verse loads and can be computed by the methods of moments just 
as though there were no end pulls. To find the stresses at any 
cross- section, we determine those due to the transverse forces 
(Lj, L,,, L,^, Rj and Rg) and those due to the longitudinal; then 
combine these stresses to get the total effect of all the applied 
forces. 

The stress due to the transverse forces consists of a shearing 
stress and a fibre stress; it will be called the flexural stress. The 
fibre stress is compressive above and tensile below. Let M denote 
the value of the bending moment at the section considered; i\ and 
6*2 the distances from the neutral axis to the highest and the low- 
est fibre in the section; and Sj and S^ the corresponding unit-fibre 
stresses due to the transverse loads. Then 

Mr,', M.', 

bj = _^ — ; and b., = — y— . 

The stress due to the end pulls is a simple tension, and it equals 
P; this is sometimes called the direct stress. Let S^^^ denote the 
unit-tension due to P, and A the area of the cross-section; then 

Both systems of loads to the left of a section between L, and 



* Ncvj'E. ]Jy "axis of a bonin " isnionnt tho liiu^ tlirou^'h tho contt^rs of 
gravity of all tho cross-sections. 



STEEXGTII OF MATEEIALS 



L,, are represented in Fig. 43, 1)\ also the stresses caused by them 
at that section. Clearly the effect of the end pulls is to increase the 



tensile stress (on the lower 
hbres) and to decrease the 
compressive stress (on the 
upper fibres) due to the flex- 
ure. Let Sp denote the total 
(resultant) unit-stress on the 
upper fibre, and S^ that on 
the lower fibre, due to all 
the forces acting on the beam. 
In combinincr the stresses 
there are two cases to con- 
sider : 



ih. 



ihiJ! 



|R = 



a 






lh_ 



Oi Oq 



1^ 



Ts, 



Oo"" -^l 



So 



c d 

Fig. 43. 

(1 ) The tlexural compressive unit-stress on the upper fibre is 
greater than the direct unit-stress; that is; Sj is greater than S^. 
The resultant stress on the upper fibre is * . 

S^ := Sj - S^ (compressive) ; 
and that on the lower fibre is - • 

S^ = S2 — Sq (tensile). 

The combined stress is as represented in Fig. 43. l\ part tensile 
and part compressive. 

(2) The flexural compressive unit-stress is less than the 
direct unit-stress; that is, S, is less than S^. Then the combined 
unit-stress on the upper fibre is 

S(, = Sq - Sj (tensile): 
and that on the lower fibre is 

S^ := S, -f So (tensile). 

The combined stress is represented by Fig. 43, c/, and is all 
tensile. 

Example. A steel bar 2x0 inches, and 12 feet long, is sub- 
jected to end pulls of 45,000 pounds. It is supported at each 
end, and sustains, as a beam, a uniform load of 6,000 pounds. 
It is required to compute the combined unit-fibre stresses. 

Evidently the dangerous section is at the middle, and M = 
|- W/; that is. 



STRENGTH OF MATERlxYLS 88 



M = -i- X (>,()()0 X 12 ^ 9,000 foot-pounds, 

8 

or ^ 9,000 X 12 =- 108,000 inch-pounds. 

The bar being placed v\ith the six-inch side vertical, 
(", = ^'2 ^^ ^ inches, and 
I = -i- X 2 X 6^ = 30 inchest (See Art. 52.) 

TT o o 108,000 X 3 .. ^^^ , . , 

Hence Sj = S,, = -— = 9,000 pounds per square inch. 

Since A=^2 X = 12 square inches, 

^ 45,000 ^^_ , . , 

Oq == — :p- — = 3,7o() pounds per square inch. 

The greatest value of the combined compressive stress is 

Sj - So ^ 9,000 - 3,750 = 5,250 pounds per square inch, 
and it occurs on the upper fibres of the middle section. The great- 
est value of the combined tensile stress is 

S, -f S„ = 9,000 4- 3,750 = 12,750 pounds per square inch, 
and it occurs on the lowest fibres of the middle section. 



EXAflPLE FOR PRACTICE. 

Change the load in the preceding illustration to one of 6,000 
pounds placed in the middle, and then sohe. 

. ( Sg = 14,250 pounds per scpiare incli. 
I ^t -- 21,750 

75. Flexure and Compression. Imagine tlie arrowheads on 
P reversed; t\wn Fig. 43, (/, will re])resent a l)eam under com- 
bined llexural and compressive stresses. The Hexiiral unit-stresses 
are com])uted as in the preceding article. The direct stress is a 
coiripression e(]ual to P, and the unit-stress dne to P is coinj)Uteil 
as in ll>e preceding articK'. Evidently tlie elfect of 1* is to iiu'i\'ase 
the com])ressive sti'ess and decrease tlie tensile stress (\\\c \o the 
flexure. In combining, we have two cases as before: 

(1) The llexni'al tensile unit-sti\'ss is gi-enter than tlu* 
direct unit-stress; that is, S, is o-reatiM" than S... Then the com- 
bined unit-stross on the lower iibre is 



84 



STBENGTPI OF MATEKIALS 



a>- 



^t = ^2 ~ ^o (tensile) ; 
and that on the upper fibre is 

^c "= ^1 + Sq (compressive). 
The combined fibre stress is represented by Fig. 44, a^ and is part 
tensile and part compressive. 

(2) The flexnral unit-stress on the lower fibre is less than 
the direct unit-stress; that is, Sg is less than S^. Then the com- 
bined unit-stress on the lower fibre is 

Sj. = Sq - S2 (compressive); 
and that on the upper fibre is 

S^j = Sq + Sj (compressive). 
The combined fibre stress is represented by 
Fig. 44, ^, and is all compressive. 

Exam/pie. A piece of timber 6x6 
inches, and 10 feet long, is subjected to end 
pushes of 9,000 pounds. It is supported in 
a horizontal position at its ends, and sustains 
a middle load of 400 pounds. Compute the 
combined fibre stresses. 

Evidently the dangerous section is at the 
middle, and M = J PZ; that is, 



Fig. 44. 



:^r = J X 400 X lo 
4 



1,000 foot-pounds, 



or 1,000 X 12 =- 12,000 inch-pounds. 

Since (\ == (\ = 3 inches, and 

1^ — Ia' = —XQX6' = 108 inches*, 

12,000 X 3 1 

fej = !^2 ^= TTjo ^= ^^^~o~ poinds per square inch. 

Since A = 6 X 6 = 36 square inches, 

^ 9,000 .,„ , . , 

o = — ^?t~^^ ^^ pounds per square inch. 
Hence the greatest value of the combined compressive stress is 
^0+^1=^ 333-iT- + 250 = 583-75- pounds per square inch. 



STKEI^GTH OF MATEEIALS 



85 



It occurs on the upper fibres of the middle section. The greatest 
value of the combined tensile stress is 

S., - S^, = '388-7^- - 250 = 83-— pounds per square inch. 

Tt occurs on the lowest fibres of the middle section. 

EXAMPLE FOR PRACTICE. 

Change the load of the preceding illustration to a uniform 
load and solve. 

j 8(3 = 417 pounds per square inch. 

( St = 83 " " " " (compression). 

76. Comhined Flexural and Direct Stress by flore Exact 
Formulas. The results in the preceding articles are only approxi- 
mately correct. Imagine the 



Ans. 





beam represented in Fig. 45, ((, 
to be first loaded with the trans- 
verse loads alone. They cause 
the beam to bend more or Jess, 
and produce certain liexural 
stresses at each section of the 
beam. Now, if end ])nlls are 
applied they tend to straighten 
the ])eam and hence diminish the flexural stresses. This effect 
of the end pnlls was omitted in the discussion of Art. 74, and 
tlie results there given are therefore only approximate, the 
value of the greatest combined fibre Tin it-stress (S^) being too 
large. On the other hand, if the end forces are pushes, they in- 
crease tlie bendino;, and therefore increase the flexural fibre stresses 
already caused by the transverse forces (see I'ig. 45, //). The 
results indicated in Art. 75 must therefore in lliis case also be 
regarded as only ap])i'().\iniate, the valne of the greatest unit- 
fibre stress (8^.) being too small. 

For beams loaded in the middle or with a uniform load, the 
followino; foi-mulas, which take into account the tlcxurnl elViH't of 
ilu) end forces, may be used : 

M denotes bendino- moment at the midtUe section of the beam; 
I denotes tlu^ monuMit of inertia of the middh^ stu'tion with 
respiH't to t]u> n(Uitral axis; 



86 STRENGTH OF MATERIALS 

Sj, S.„ <\ and c^ have the same meanings as in Arts. 74: and 
75, but refer always to the middle section ; 

I denotes length of the beam; 

E is a number depending on the stiffness of the material, the 
average values of which are, for timber, 1,500,000; and for struc- 
tural steel 30,000,000.* 

Q Me. . t. Me,, 

bj=- 1. . , ana b 



lOE lOE 

The plus sign is to be used when the end forces P are pulls, and 
the minus sign when they are pushes. ' 

It must be remembered that S^ and S, are flexural unit- 
stresses. The combination of these and the direct unit-stress is 
made exactly as in articles 74 and 75. 

ExamiJfles. 1. It is required to apply the formulas of this 
article to the example of article 74. 

As explained in the exaraple referred to, M = 108,000 inch- 
poiinds; 6'^= e^— 3 inches; and I = 36 inchest 
Now, since Z = 12 feet = 144 inches, 
e c. 108,000X3 324,000 

^^ = ^^ = ; 45,000 X 144^ - 361^3X1 = ^-'^^ I"^^^"^^ 

^^' "^ 10 X 30,000,000 
per square inch, as compared with 9,000 pounds per square inch, 
the result reached by the use of the approximate formula. 
As before, S^ = 3,750 pounds per square inch; hence 

S^ = 8,284- 3,750 = 4,534 pounds per square inch; 
and St =- 8,284 + 3,750 = 12,034 " " " " ' 

2. It is required to apply the formulas of this article to the 
example of article 75. 

As explained in that example, 

M == 12,000 inch-pounds; 
L\ = c, =- 3 inches, and I = 108 inchest 
Now, since J ^= 120 inches, 

12,000 X 3 36,000 .^^^ , 

^- = ^- - 7;;7 ~^ Q00 X 120- = 108 - 8.64 = ''^ T^^^^^^ 
10 X 1,500,000 



* Kgte. This quantity " E " is more fully explained in Article 95. 



STEEXGTir OF MATEEIALS 87 

per s(juare inch, as coiii[)arecl with •^•^•3.'j pounds per square inch, 
the result reached by use of the approximate method. 
As before, 8^ =^ 2'5() pounds per square inch; hence 

Sc = 3G2 -\- 250 --= ()12 pounds per square inch; and 
St = 802 - 250 = 112 '- - " " . 

EXAMPLES FOR PRACTICE. 

1. Solve the example for practice of Art. 71 by tlic foi-niulas 

of this article. 

\ , j ^c "^^ 12,820 ])ounds T)er scpiare inch. 
^^"''' / St =- 20,}32() '• 

2. Solve the example for practice of Art. 75 by the formulas 
of this article. 

\ . j ^c -= ^'^^^ pounds per square inch. 

/ Sj =3 70 '. a 'i a (compression"). 

STRENGTH OF COLUHNS. 

A stick of timl)er, a bar of iron, etc., when used to sustain 
end loads which act len(ythwise of the pieces, are called columns, 
posts, or struts if they are so loner that they would bend before 
breakino-. When they are so sliort that they would not l)end 
))efoj-e breaivint^, they are called short blocks, and tlieir comj)res- 
sive streno;ths are computed by means of e(piation 1. The strencrths 
of cohimiis cannot, however, be so simply determiiunl. and we now 
proceed to explain the method of computino- them. 

77. End Conditions. The strenoth of a eolumii dej)ends in 
])art on the way in which its ends bear, or are joined to other 
])arts of a struct nre, that is, on its ''end conditions." 'J'here are 
j)i"actically l)iit tlii'ee kinds of end conditions, namely: 

1. "Hin^v/'or "pin" onds, 

2. ".I'"'lat'' ()!• '• s(|iiMr(i " ends, and 
'.]. '• Fixed " ends. 

(1 ) AVhen a column is fastened to its suj)j)ort at one end by 
means of a pin about which the column could rolali' if the other 
end were free, it is said to be .** liiuo'ed " or '* piinuMl "" at the 
former end. I>rido;e posts or columns are often hinged at the ends. 

(2) A column cilhei- end of wliit-h is llat and pei-pendieulai- 
to its axis and bears on olhei- parts of the struetui'e at that sui-faee, 
is said to be " lint " oi" '* sipiai'e"* at th.'il end. 



S8 



STREXGTH OF MATERIALS 



I 3 ) L oiiiuius are sometimes riveted near their ends directly 
to other parts of the strnctnre and do not bear directly on their 
ends; such are called •• fixed ended/' A colnmn which bears on its 
dat ends is often fastened near the ends to other parts of the struc- 
ture, and such an end is also said to l)e " fixed." The fixincr of an 
end of a column stiffens and therefore strengthens it more or less, 
but the strength of a column with fixed ends is computed as 
though its ends were flat. Accordingly we have, so far as strength 
is concerned, the followincr classes of columns : 

78. Classes 01 Columns. 1 ) Both ends hinged or pinned; 
(2) one end hinged and one flat; (3) both ends flat. 

Other thincrs beincr the same, columns of these three classes 
are unequal in strength. Columns of the first class are the 
weakest, and those of the third class are the stroncrest. 



A 


r 


A 









B 

6w 







B 










A 


. 


A 




1 








B 





Yis. i6. 



79. Cross=sections of Columns. Wooden columns are usu- 
ally solid, square, rectanoular, or round in section; but sometimes 
they are '* built up '' hollow. Cast-iron columns are practically 
always made hollow, and rectancrular or round in section. Steel 
columns are made of sino-le rolled shapes — angles, zees, channels, 
etc. ; but the larger ones are usually " built up '' of several shapes. 
Fig. 46, e/, for example, represents a cross -section of a ••Z-bar'* 
column: and Ficf. 46. '/, that of a '-channel'' column. 

80. Radius of Gyration. There is a quantity appearing in 
almost all lormiiias fur riie strencrth of columns, which is called 
•• radius of gyration." It depends on the form and extent of the 
cross-section of the column, and may l)e defined as follows : 



STKEXGTH OF MATEEIALS 89 

The radius of gyration of any plane figure (as the section of a column) 
with respect to any line, is such a length that the square of this length mul- 
tiplied by the area of the figure equals the moment of inertia of the figure 
with respect to the given line. 

Thus, if A denotes the area of a figure; T, its moment of in- 
ertia with respect to some line; and r, the radiuc of gyration 
with respect to that line; then 

r'K =- I; or r = Vl -^ A. (p) 

In the column formulas, the radius of gyration always refers to ai\ 
axis through the center of gravity of the cross-section, and usually 
to that axis with respect to which the radius of gyration (and mo- 
ment of inertia) is least. (For an exception, see example 3. 
Art. S3.) Hence the radius of gyration in this connection is often 
called for brevity the ''least radius of gyration," or simply the 
" least radius." 

Examples. 1. Show that the value of the radius of o-yration 
given for the square in Table A, page 54, is correct. 

The moment of inertia of the square with respect to the axis 
is -i2^(^. Since A = «-, then, by formula 9 ^bove. 



1 ,. I 1 



2. Prove that the value of the radius of gyration criven for 
the hollow s(piare in Table A, page 54, is correct. 

The value of the moment of inertia of the s(piare with respect 
to the axis is ^'., [(/^ - (f^^). Since A = (r - (/ -', 



\ {"' - ">') 



r =z: 



M^v -w -J 



EXAHPLE FOR <PRACTICE. 

Prov(^ that the values of the radii of oyration of the other fin-- 
ures given in Table A, pag(^ 51, are corriM*!. The axis in each 
case is indicated by tli(^ line throuoh the cenliM* of <'-ravitv. 

8i. Radius of Gyration of lUiill up Sections. Iln' ladius of 
gyration of a bnilt-ujt section is eonipntcd siniilarU to that iA' any 
other ligni'e. i''ii-sl, we ha\(' to ('onij)Ute the nionient o\' iru^rtia of 



90 STRENGTH OF MATERIALS 

tLe section, as explained in Art. 54; and then we use formula 9, as 
iii the examples of the preceding article. 

ExamjjJe. It is required to compute the radius of gyration 
of the section represented in Fig. 30 (page 52) with respect to the 
axis AA. 

In example 1, Art, 54, it is shown that the moment of inertia 
of the section with respect to the axis AA is -129 inchest The 
area of the whole section is 

2 X 0.03 -j- T = 19.00; 

hence the radius of gyration /'is 



/= = a;— = 4.74 inches. 

\ 19.00 

EXAMPLE FOR PRACTICE. 

Compute the radii of gyration of the section represented in 
Fig. 31, a, with respect to the axes AA and BB. (See examples 

for practice 1 and 2, Art. 54, ) 

. \ 2.87 inches. 

Ans. 

/ 2.09 •• 

82. Kinds of Column Loads. AVhen the loads applied to a 
column are such that their resultant acts throuo-h the center of 
gravity of the top section and along the axis of the column, the 
column is said to be centrally loaded. When the resultant of the 
loads does not act through the center of gravity of the top 
section, the column is said to be eccentrically Soaded. All the 
following formulas refer to columns centrally loaded. 

83. Rankine's Column Formula. AVhen a perfectly straight 
column is centrally loaded, then, if the column does not bend and 
if it is homogeneous, the stress on every cross-section is a uniform 
compression. If P denotes the load and A the area of the cross- 
section, the value of the unit-cOmpression is P ^- A. 

On account of lack of straio-htness or lack of uniformity in 
material, or failure to secure exact central a])j)lication of the load, 
the load P has what is known as an '' arm '' or '' leverage '^ and 
bends the column more or less. There is therefore in such a 
column a bending; or flexural stress in addition to the direct com- 
pressive stress above mentioned; this bending stress is com])ressive 



STKENGTH OF MATEKIALS 



91 



on the concave side and tensile on the convex. The value of the 
stress per unit-area (unit-stress) on the fibre at the concave side, 
according to equation 0, is Mc -^ I, where M denotes the bending 
moment at the section (due to the load on the column), c the 
distance from the neutral axis to the concave side, and I the 
moment of inertia of the cross-section witli respect to the neutral 
axis. fXotice that this axis is perpendicular to the j)lane in 
which the column bends.) 

The upper set of arrows (Fig. 4T) represents the direct com- 
pressive stress; and the second set the bending stress if the load 
is not excessive, so that the stresses are within the elastic limit of 
the material. The third set represents the combined stress that 
actually exists on the cross-section. The greatest combined unit- 
stress evidently occurs on the fibre at the concave side and where 

the deflection of the column is greatest. The 
stress is compressive, and its value S per unit- 
area is given by the formula, 

P Mr 




S 



^jjjrrn — ■Mllljj 



ynrrrTTTTTTTTT 



yrrn 



i'^iir. 47. 



Xow, the bending moment at the place of 
greatest deflection equals the product of the 
load P and its arm (that is, the deflection). 
Calling the deflection ^/, we have M = P//; and 
this value of M, substituted in the last equa 
tion, gives 

s„..i' , ■'/'■ 



Let r denote tlit^ radius of gyration of the cross-section n\ ilh rrspce 
to the neutral axis. Then I ^^ A)-^ (see equation U); and this 
value, substituted in tlu^ last e(]uation, gives 

P Fdc 



S = 



A 



A/ 



.».2 



P >/r 

A ^ I' 



AccordiiH'' lo the ibcoiv of (lie stillness of beams on end sun- 
ports, deflect ions vary dircclly as the s(piare of the lengtli /, and in 
versely as the distance '- fi'om the nculral axis lo ll\c reinoti'st til>rc 
of tile cross-section. Assuinini'' tiial the dcllcctit)ns of columns 



92 STKENGTII OF MATERIALS 

follow the same laws, we may write d = Jc (/- -^ c), where Z." is 
some constant depending on the material of the column and on the 
end conditions. Substituting this value for d in the last equation, 
we find that 

: p T' ^ ' 



and 



p 


S 




A 
P — 


1 + Z;- 
SA 






1 + Z; 


9 



(lO) 



Each of these (usually the last) is known as *' Rankine's formula." 

For mild-steel columns a certain large steel company uses S = 50,000 
pounds per square inch, and the following values of J: : 

1. Columns with two pin ends, l' = 1-^ 18,000. 

2. " " one flat and one pin end, k = 1 -^ 24,000. 

3. " " both ends flat, k =1-^ 36,000. 

With these values of S and Jc, P of the formula means the ultimate load, 
that is, the load causing failure. The safe load equals P divided bv the 
selected factor of safety — a factor of 4 for steady loads, and 5 for moving 
loads, being recommended by the company referred to. The same unit is to 
be used for I and r. 

Cast-iron columns are practically always made hollow with 

comparatively thin walls, and are usually circular or rectangular 

in cross-section. Th" followincr modifications of Pankine's formula 

are sometimes used: 

p 80.000 1 

For circular sections. 



A -, , l^ 



For rectangular sections. 



' 800 d^' 1 
p _ 80.000 [" (lO') 

A ". , I' i 



1.000 d^ 



J 



In these formulas d denotes the outside diameter of the circular sec- 
tions or the length of the lesser side of the rectangular sections. The same 
unit is to be used for I and d. 

Examjples. 1. A 40-pound 10-inch steel I-beam 8 feet 
hmg is used as a fiat-ended column. Its load being 100,000 
pounds, what is its factor of safety ? 

Obviously the column tends to bend in a plane perpendicular 
to its web. Hence the radius of gyration to be used is the one 



STKENGTH OF MxVTERIxVLS 03 



with respect to that central axis of tlie cross-section which is in 
the web, that is, axis 2-2 (see figure accompanying table, page 72 j . 
The moment of inertia of the section with respect to that axis, 
according to the table, is 9.50 inches^; and since the area of the 
section is 11.70 square inches, 

IS^oWy I = H feet = 90 inches; and since /i; = 1 -~ 30,000, and S = 
50,000, the breaking load for this column, according to llankine's 
formula, is 

p 50,000 X 11.76 .u'-7(iA 1 

r* = 1 -— r=z 446,790 pounds. 

^ + 30,000 X 0.81 

Since the factor of safety ecpials the ratio of the breaking h)ad to 
the actual load on the column, the factor of safety in this case is 

446,790 ... 

2. What is the safe load for a cast-iron column 10 feet lono- 
with square ends and a hollow rectangular section, the outside 
dimensions being 5x8 inches; the inner, 4x7 inches; and the 
factor of safety, 6 ? 

In this case I ^ -- 10 feet = 120 inches; A ^^. 5x8-4x7 
= 12 square inches; and d =^-- 5 inches. lliMice, according to 
formula 10', for rectaui^uhu sections, the breakino- load is 

80,000 X 12 ., , ^ 

P z. . '- :r-y-^ "- ()10,0()0 pOUnds. 

1 -I- ^:^ 

' 1,000 X 5^ 

Since the safe load cMiiials the l)reakiii<>- load di\ ided b\ tin- factor 
of safety, in this case the safe load eipials 

(;l(),o()o 

__ ' 101,700 iM)iiii.ls. 

3. iV channel colunin (see 1^'ig. 40, //) is pin-ended, the |)in^ 
being jjerpeiidiculai- to tlic webs of the channel ( ri'j)resented b\ 
A A in the lioii I'c ) , and its K'noth is 1() h'ct (dislanci* bftweiMi axes 



94 stee:n"gtii of materials 

of the pins). If the sectional area is 23.5 square inches, and the 
moment of inertia with respect to AA is 386 inches* and with 
respect to BB 214 inches^ what is the safe load with a factor of 
safety of 4 ? 

The column is liable to bend in one of two ways, namely, in 
the plane perpendicular to the axes of the two pins, or in the plane 
containing those axes. 

( 1) For bending in the first plane, the strength of the col- 
umn is to be computed from the formula for a pin -ended column. 
Hence, for this case, i^- = 386 ~- 23.5 = 16; and the breaking 
load is 

50,000 X 23.5 , ^ ,, , , - 

-7zrT> -, ^ o = 1,041,600 pounds. 

(lb X 12)- - ^ 



1 + 



18,000 X 16 



The safe load for this case equals -- — j^ = 260,400 pounds, 

(2) If the supports of the pins are rigid, then the pins 
stiffen the column as to bending in the plane of their axes, and the 
strength of the column for bending in that plane should be com- 
puted from the formula for the strength of columns with flat ends. 
Hence, 'i-' = 214 -f- 23.5 =: 9.11, and thebreakino- load is 

50,000 X 23.5 

1 = ,-,,> ,^ -,.v,9 = 1,056,000 pounds. 

(16 X 12; ^ 

"^ 36,000 X 9.11 

l,0e56,000 ^., ,r. 
The safe load for this case equals j = 264,000 pounds. 



EXAMPLES FOR PRACTICE. 

1. A 40-pound 12-inch steel I-beam 10 feet long is used as 

a column with flat ends sustaining a load of 100,000 pounds. 

AVhat is its factor of safety? 

Ans. 4.1 

2. A cast-iron column 15 feet long sustains a load of 
150,000 jDOunrls. Its section ])eing a hollow circh\ 9 ijiclies out- 
side and 7 inches inside diameter, what is the factor of safety^ 

Ans! 8.9 

3. A steel Z-bar column (see Fig. 46, a) is 24 feet long and 
has square ends; the least radius of gyration of its cross-section is 



STEENGTH OF MATEEIALS 



05 



3.1 inches; and tlie area of the cross-section is 24. 5 square inches. 
What is the safe load for the column with a factor of safety of i i 

Ans. 247,000 pounds. 

4. A cast-iron column 13 feet lono- has a hollow circular 
cross-section 7 inches outside and 5A- inches inside diameter. 
What is its safe load with a factor of safety of G? 

Ans. 121,142 pounds. 

5. Compute the safe load for a 40-pound 12-inch steel 
I-beam used as a column with flat ends, its lencrth heina 17 feet. 

Use a factor of safety of 5. 

Ans. 52.4:70 pounds. 

84. Graphical Representation of Column Formulas. Col 

umn (and most other engineering) formulas can be represented 
graphically. To represeiit Eankine's formula for flat-ended mild- 
steel columns, 

P 50,000 



1 + 



36,000 



wo first substitute different values of I -^ /' in the formula, and 
solve for P -^- A. Thus we find, when 

l^r = 40, P - A = 47,9(K) ; 
l^r= 80, P - A = 42,500 ; 
Z - r = 120, P - A = 35,750 ; 
etc., etc. 

Now, if these values of I -^ r be laid off by some scale on a line 
from (), Kig. 48, and the corres])onding values of I* -'- A be laid 




lOO 



Fip. 48. 



20O 



300 



off Ncrlicall V fi-oiii \\\k\ points on the line, we get a series of j)oints 
>is (I ^ />, (\ etc.; and a sni()t)tli eurvi' through the j)oints </, A, r, 



96 STKE:N"GTH of IMATEPJALS 

etc., represents the formula. Such a curve, besides representing 
the formula to one's eye, can be used for lindinop tlie value of 
F -^ A for any value oil -^ r; or the value of / -f- ;' for any value 
of F -^ A. The use herein made is in explaining other column 
formulas in succeeding; articles. 

85. Combination Column Formulas. Many columns have 
been tested to destruction in order to discover in a practical way 
the laws relatino- to the streno-th of columns of different kinds. 
The results of such tests can be most satisfactorily represented 
graphically by plotting a point in a diagram for each test. Thus, 
suppose that a column whose I -^ r was 80 failed under a load of 
276,000 pounds, and that the area of its cross-section was 7.12 
square inches. This test would be represented by laying off Oc?, 
Fig. 49, equal to 80, according to some scale; and then ah equal to 
276,000 H- 7.12 (F -^ A), according to some other convenient 
scale. The point h would then represent the result of this par- 
ticular test. All the dots in the figure represent the way in which 
the results of a series of tests appear when plotted. 

It will be observed at once that the dots do not fall upon any 
one curve, as the curve of Ilankine's formula. Straio-ht lines and 

50000- 



AOOOO-:^ 
30000 ■- 

20000 -- 
10000•- 



b 



I 



aoo 200 300 

Fig. 49. 

curves simpler than the curve of Rankine's formula have been 
fitted to represent the average positions of the dots as determined 
by actual tests, and the formulas corres])onding to such lines have 
been deduced as column formulas. These are explained in the 
followincr articles. 

86. Straight=Line and Euler Formulas. It occurred to Mr. 
T. II. Johnson that most of the dots corresponding to ordinary 





I 



I 




C/3 
C/) 

M 
CC 
H 

o 2 

S "< 

Q ■:; 

a: <; 

OQ ^ 

K - 

W ~ 

Q ? 

z ^ 

= V 

Q -r- 

O ^ 

O i 

^ =^ 

If) '^ 

g s 

K -^ 

> ? 

o -^ 

w 5 

K - 

S ■; 

h : 

<^ S 

5 "^ 

o "^ 
Q 



STRENGTH OF MATERIALS 



97 



lengths of columns agree with a straight line just as well as with 
a curve. He therefore, in 1880, made a number of such plats or 
diagrams as Fig. 49, fitted straight lines to them, and deduced the 
formula corresponding to each line. These have become known 
as "straight-line formulas," and their general form is as follows: 



— T- = b — 'III 5 

A /' 



(") 



P, A, Z, and /'having meanings as in Rankine's formula (Art. 83), 
and S and in beino; constants whose values accordincr to Johnson 
are given in Table E below\ 

For the slender columns, another formula (Euler's, long since 
deduced) -was used by Johnson. Its general form is — 

A = (TT-^^ ('^) 

n beinof a constant whose values, according; to Johnson, are given 
in the followino- table:. 

o 

TABLE E. 

Data for Mild=^Steel Columns. 





S 


m 


Limit {l -i- r) 


n 


Hinged onds 

Flat ends 


52,500 
52,500 


220 
180 


IGO 
195 


441XKK),(K)0 
600,000,1 KK) 







The numbers in the fourth column of the table mark the point of divi- 
sion between columns of ordinary length and slender columns. For the 
former kind, the straight-line formula applies: and for the second, Euler's. 
That is, if the ratio I ^ r for a steel column with hinged end, for example, is 
less than 160, we must use the straight-line formula to compute its safe load, 
factor of safety, etc.; but if the ratio is greater than 100, we must use Euler's 
formula. 

For cast'ir^on column s with flat ends, S = 34,(XX1, and in = SS: and since 
they should never be used "slender," there is no use of Euler's formula for 
cast iron columns. 

The line AB, Fig. 50, represents Johnson's straight-line for- 
mula; and BC, Euler's forimila. It will be noticed that the two 
lines are tangent; the point of tangiMU'v corresponds to the •' lini- 
iting" value" / -:- v, as indicated in the table. 

/'J.i'tfi/tji/t's. I. A lO-ponnd Kl-ineh .Ueel 1 beam eoluinn 8 



98 



STRENGTH OF MATERIALS 



feet long sustains a load of 100,000 pounds, and the ends are iiat. 
Compute its factor of safety according to the methods of this 
article. 

The first thing to do is to compute the ratio I -^ r for the 
column, to ascertain whether the straight-line formula or Euler's 




lOO 



200 



300 



Fig. 50. 

formula should be used. From Table C, on page 72, we find that 
the moment of inertia of the column about the neutral axis of 
its cross-section is 1).50 inches*, and the area of the section is 
11.76 square inches. Hence 

9.50 



r' = 



Since I 



11.70 

8 feet = 90 inches 
I 96 



= O.Sl; ovr 



0.9 inch. 



r 



0.9 = 106^ 



This value of ^ -^ ^' is less than the limiting value (195) indicated 
by the table for steel columns with flat ends (Table E, p. 97j, and 
we should therefore use the straight-line formula; hence 



11.76 



7 = 52,500 



180 X 106-^' 



or. 



11.76 (52,500- 180 X 106^;:^) == 391,600 pounds. 



This is the breaking load for the column according to the straight- 
line formula; hence the factor of safety is 



391,600 ^ g g 



100,000 



STKEl^GTH OF MATEEIALS 99 

2. Suppose that the length of the colunin desci-ibed in the 
preceding example were 16 feet. What would its factor of safety be? 

Since I = IC) feet = 192 inches; and, as before, '/■ = 0.9 
inch, I -^ r = 21;3J. This value is greater than the limiting 
value (195) indicated by Table E (p. 97) for fiat-ended steel col- 
umns; hence Euler's formula is to be used. Thus 

P 060,000,000 



11.70- (218^/ ' 

11.70 X 000,000,000 
or, 1 = ,,>-,., TV., = 1< 2,100 pounds. 

This is the breaking load; hence the factor of safety is 

172,100 _ ^ ^ 

100,000 ~~ 

3. What is the safe load for a cast-iron column 10 feet loner 
with square ends and hollow rectangular section, the outside 
dimensions being 5x8 inches and the inside 4X 7 inches, with a 
factor of safety of ? 

Substituting in the formula for the radius of gyration given 
in Table A, page 54, we get 



8 X 5^ - 7 X 4:^' . (-^ . , 

^ l.Du inches. 



12 (8 X 5-7 X 4:j 
Since I = 10 feet = 120 inches, 

According to the straight-line formula for cast iron, A being 
equal to 12 square inches, 

— = 34,000-88 X 61.22; 

JL isj 

or, P -= 12 (34,000 - 8S x 01.22) --. 343,360 pounds. 

This being the breaking load, the safe load is 

343,360 




LOFC. 



-= 57,227 pounds. 



100 STEEXGTH OF MATEKIALS 

EXAMPLES FOR PRACTICE. 

1. A 4:0-ponnd 12-inch steel I-beam 10 feet long; is used as 
a flat-ended column. Its load being 100,000 pounds, compute 
the factor of safety by the formulas of this article. 

Ans. 3.5 

2. A cast-iron column 15 feet long- sustains a load of 
150,000 pounds. Its section being a hollow circle of 9 inches 
outside and T inches inside dir.meter, compute the factor of safety 
by the straight-line formula. 

Ans. 4.8 

3. A steel Z-bar column (see Fig. 46, a) is 24 feet long 
and has square ends; the least radius of gyration of its cross- 
section is 3.1 inches; and the area of the cross-section is 24.5 
square inches. Compute the safe load for the column by the 
formulas of this article, using a factor of safety of 4. 

Ans. 219,000 pounds. 

4. A hollow cast-iron column 13 feet long has a circular 

cross-section, and is 7 inches outside and 5-| inches inside in 

diameter. Compute its safe load by the formulas of this article, 

using a factor of safety of 6. 

Ans. 68,500 pounds 

5. Compute by the methods of this article the safe load for 
a 40-pound 12 -inch steel I-beam used as a column with flat ends, 
it the lencfth is 17 feet -and the factor of safety 5. 

Ans. 35,100 pounds. 

87. Parabola=EuIer Formulas. As better fittino- the results 

of tests of the strength of columns of '• ordinary lengths," Prof. 

J. B. Johnson proposed (1892) to use parabolas instead of straight 

lines. The general form of the ''parabola formula" is 

P 7 

_ = S-7«(_j-, (13) 

P, A, I and /■ hayincr the same meanings as in Pankine's formula. 
Art. 83; and S and iji denotinor constants wliose yalues, according^ 
to Professor Johnson, are giyen in Table F below. 

Like the straight-line formula, the parabola formula should 
not be used for slender columns, but the following (Euler's) is 
applicable; 



STRENGTH OF MATERIALS 



101 



1 n 

the values of ih (Johnson j l)eino^ oriven in the following table: 

TABLE F. 
Data for flild Steel Columns. 





S 


m 


Limit {I -h ?•) 


n 


Hin<4od ends 

Flat ends 


• 42,000 
42,CXJ0 


0.97 
0.62 


150 
190 


450,(XXUJ(K) 



The point of division bet'ueen columns of ordinary length and slender 
columns is given in the fourth column of the table. That is, if the ratio l^r 
for a column with hinged ends, for example, is less than 150, the parabola 
formula should be used to compute the safe load, factor of safety, etc.; but 
if the ratio is greater than 150, then Euler's formula should be used. 

The line AB, Fig. 51, represents the parabola formula; and the line 
BC, Euler's formula. The two lines are tangent, and the point of tangency 
corresponds to the ' ' limiting value ' ' l^r of the table. 

For wooden columns square in cross-section, it is convenient to replace 
r by d, the latter denoting the length of the sides of the square. The formula 
becomes 

= S - m (^)^ 



A 



d 



S and fii for flat-ended columns of various kinds of wood having the follow- 
ing values according to Professor Johnson: 

For White pine, S=2,500, w = 0.6; 

" 8hortdeaf yellow pine, S=3,300, 731 = 0.7; 

'' Longdeaf yellow pine, 8=4,000, 7n = 0.8; 

" White oak, 8=3,500, m = O.S. 

The preceding formula applies to any Avooden column whose ratio. I-^d, 

is less than 60, within which limit columns of practice are included. 




20000 -- 



loooo-■ 



300 



E'.vcnn ph.'i. 1. A lO-poniid 10 incdi sleid 1 Ix'am eolninn 



102 STEE^X^TH OF MATERIALS 

S feet long sustains a load of 100,000 pounds, and its ends are tiat. 
Compute its factor of safety according to tlie methods of this 
article. 

The first thing to do is to conipnte the ratio 1 -^ i" for the 
column, to ascertain -whether the parabola formula or Euler's for- 
mula should be used. As shown in exanijile 1 of the preceding 
article, 1 -^ r =^ lOO^. This ratio beino- less than the limitincr 

-on o 

value, 190, of the table, we should use the parabola formula. 
Hence, since the area of the cross-section is 11.70 square inches 
(see Table C, page 72 ), 

-L_ = 42,000 - 0.02 fl06|V-': 



11.76 

or, . P = 11.76 [42,000 - 0.62 (106f )'] = 110,970 pounds. 

This is the breakino- load accordino- to the parabola formula: hence 
the factor of safety is 

410,970 _ ^^ 

100,000 

2. A white pine column 10 X 10 inches in cross-section and 
18 feet long sustains a load of 40,000 pounds. TThat is its fac4;or 
of safety ? 

The lengrth is IS feet or 21i] inches; hence the ratio / h- <:Z = 
21.6, and the parabola formula is to be applied. 
Xow, since A = 10 x 10 = lUO square inches, 

P 

j^-= 2,500-0.6 X 21.6=; 

or, P = 100 (2,500 - 0.6 X 21.6-) = 222,000 pounds. 

This beino' the brieakino; load accordino- to the r)arabola formula, 
the factor of safety is 

222,000 
40,000 "" ^"^ 

3. "What is the safe load for a long-leaf yellow pine column 
12 X 12 inches square and 30 feet lono-. the factor of safety 
beino- 5 ? 

The leng-th beino- 30 feet or 300 inches, 

- = ^ - SO- 
d 12 - ''^' 



STRENGTH OF MATERIALS 10:i 

hence the parabola formula should be used. Since A = 12 X 12 
= 144 square inches, 

P 

~~r = 4,000 - 0.8 X 30^: 
144 ' ' 

or, P = 144 (4,000 - 0.8 X 30^) = 472,320 pounds. 

This being the breaking load according to the parabola formula, 

the safe load is 

472,320 

^ = 94,46o pounds. 

EXAMPLES FOR PRACTICE. 

1. A 40-pound 12-inch steel I-beam 10 feet long is used as 
a flat-ended column. Its load being 100,000 pounds, compute its 
factor of safety by the formulas of this article. 

Ans. 3.8 

2. A white oak column 15 feet lontr sustains a load of 

30,000 pounds. Its section being 8x8 inches, compute the 

factor of safety by the parabola formula. 

. Ans. 0.0 

3. A steel Z-bar column (see Fig. 40, a) is 24 feet long and 
has square ends; the least radius of gyration of its cross-section 
is 3.1 inches; and the area of its cross-section is 24.5 scpiare 
inches. Compute the safe load for the column by the formulas 
of this article, using a factor of safety of 4. 

Ans. 224,500 ])()uiids. 
4.- A short-leaf yellow ])ine column 14 X 14 inches in sec- 
tion is 20 feet lono;. AVhat load can it sustain, with a factor of 

safety of (> ? 

Ans. 101,000 ]H)Uuds. 

88. *' Broken Straight=Llne " Formula. A large steel com- 
pany. com])ntes the strength of its ilat-eiided steel columns by two 
formulas ]-epresented by two straight lines Al> and 1>(\ Fig. 52. 
The formulas are 

p / 

and -^ =-- 08,400 - 228 - , 

A r 

]*, A, /, and /' lia\ inc tlie same meanint's ms in Art. S)^. 



104 



STEEXGTH OF MATEKIALS 



The point B corresponds very nearly to the ratio I -^ r = UO. 
Hence, for columns for which the ratio / -^ r is less than 90, the 
first formula applies; and for columns for which the ratio is 
greater than 90, the second one applies. The point C corre- 
sponds to the ratio I -^ r = 200, and the second formula does not 
apply to a column for which / ~ r is greater than that limit. 

50O0O 



-40000" 
3OOO0 
20000-- 
lOOOO "- 




\-7-r 



lOO 

Fiff. 52. 



200 



The ratio I ^ r for steel columns of practice rarely exceeds 150, 
and is usually less than 100. 

Ficr. 53 is a combination of Figrs. 49, 50, 51 and 52, and 
represents graphically a comparison of the Rankine, straight-line, 
Euler, parabola-Euler, and broken straight-line formulas for flat- 
ended mild-steel columns. It well illustrates the fact that our 
knowledge of the strength of columns is not so exact as that, for 
example, of the strength of beams. 

P-^A 

5CX)00^ 




lOO 



200 



300 



Fi^. 53. 



89. Design of Columns. All the preceding examples relat- 
ing to columns were on either (1) computing the factor of safety 



STREXGTH OF MATERIALS 105 

of a given loaded column, or (2) computing the safe load for a 
given column. A more important problem is to design a column 
to sustain a given load under given conditions. A complete dis- 
cussion of this problem is given in a later paper on design. AVe 
show here merely how to compute the dhnensions of the cross- 
section of the column after the form of the cross-section has been 
decided u])on. 

In only a few cases can the dimensions be compui;ed directly 
(see example 1 following), but usually, when a column formula is 
applied to a certain case, there will be two unknown quantities in 
it, A and r or d. Such cases can best be solved by trial (see 
examples 2 and 3 below). 

EQmn2)le. 1. What is the proper size of white pine column 
to sustain a load of 80,000 pounds with a factor of safety of 5, 
when the length of the column is 22 feet 'i 

We use the parabola formula (equation 13). Since the safe 
load is 80,000 pounds and the factor of safety is 5, the breaking 
load F is 

80,000 X 5 = 400,000 pounds. 

The unknown side of the (square) cross-section being denoted by 
r/, the area A is d'. Hence, substituting in the formula, since I 
= 22 feet =^r 2G4 inches, we have 



400,000 ^_^ nr^^^' 
_-^ = 2,.00-0.r>_-. 

JMultinlyinii" l)()tli sides l)y (P o'ives 

400,000 -- 2,500 d' - O.C) X 2('>4% 
or 2,500 r/2 =.. 400,000 + 0.0 x 2()42 = 441,817.6. 

Hence d' = 176.73, or d = 13.8 inches. 

2. AVhat size of cast-iron colnnm is needed to sustain a load 
of lOO.OOO ponnds willi a factor of safety of 10. tlie length of the 
colnmn beino- 14 feet 'i 

Wii sliall suj)j)Ose that it has been (U'ciiUnl to niaki' the cross- 
section circular, and shall conipnte bv Kankiiu^'s formula nioditied 
for cast-iron eolnnins (iMpiation 10'), Tlu^ br^'akin*;" load for the 
column would Ite 



106 STRENGTH OF MATEEIALS 



100,000 X 10 = 1,000,000 pounds. 
The length is 14 feet or 168 inches; hence the formula oecomes 

1,000,000 _ 80,000 ^ 

A ■ 168^' 

"^800^2 

or, reducing by dividing both sides of the equation by 10,000, and 
then clearing; of fractions, we have 

1682 



100 



[1 + sm^ = '''' 



There are two unknown quantities in this equation, d and A, and 
we cannot solve directly for them. Probably the best way to pro- 
ceed is to assume or guess at a practical value of d, then solve for 
A, and finally compute the thickness or inner diameter. Thus, let 
us try d equal to 7 inches, first solving the equation for A as far 
as possible. Dividing both sides by 8 we have 

100 p 168^ 1 

and, combining, 

441 

A = 12.5 + -^. 

Now, substituting 7 for d, we have 

441 
A = 12.5 + -Tjr = 21.5 square inches. 

The area of a hollow circle whose outer and inner diameters are 
d and d^ respectively, is 0.7854 ((P - d^^). Hence, to find the inner 
diameter of the column, we substitute 7 for d in the last expres- 
sion, equate it to the value of A just found, and solve for d^. Thus, 

0.7854 (49-^/) = 21.5- 



hence 



*^-^^>' = m = 2^-3^; 



and d^^ = 49 - 27.37 = 21.63 or d^ = 4.65. 

This value of d makes the thickness equal to 

I (7-4.65) == 1.175 inches, 



STRENGTH OF MATERIALS 



107 



which is safe. It might be advisable in an actual case to try 
<l equal to 8 repeating the computation.* 

EXAMPLE FOR PRACTICE. 

1. What size of white oak column is needed to sustain a load 
of 45,000 pounds with a factor of safety of (), the length of the 
column being 12 feet. 

Ans. d = S-J, practically a 10 X 10-inch section 

STRENGTH OF SHAFTS. 

A shaft is a part of a machine or system of jnachines, and is 
used to transmit power by virtue of its torsional strength, or resist- 
ance to twisting. Shafts are almost always made of metal and arc 
usually circular in cross-section, being sometimes made hollow. 

90. Twisting Moment. Let AF, Fig. 51, represent a shaft 
with four pulleys on it. Suppose that D is the driving puHcy 
and that B, and E are ])ulleys from which power is taken off to 
drive machines. The portions of the shafts between the pulleys 




Fitr. 54. 



are twisted when it is transmilling power; and l>y tlie twisting 
moment at any cross-section of the shaft is meant the algebraic 
sum of the moments of all the forces aetini"- on llie sliaft on either 



* NoTK. Tlio sirui'tmnl stcol liiiiulb(H)Us ctnitaiu oxtoiisivo tiibU's l>y 
iiiOHiis of which the (losif^jn of coiunins of stot^l or cast iron is mucli facililattMl. 
The (lifricultics ciicoiiiilci-cd in (ho use of fonnulio :iro woll ilhist ratod in thi> 
oxaini>!e. 



108 STEEXGTH OF MATERIALS 

side of the section, the moments being taken with respect to the 
axis of the shaft. Thns, if the forces actino- on the shaft ('at the 
pulleys) are P^. P.. P.. and P^ as shown, and if the arms of the 
forces or radii of the pulleys are a^. cf..,^ a^^ and (T/^ respectively, then 

the twisting moment at any section in 



BC is P, a,. 
CD is Pj a, -f P 



Cf. 



D£ is P, a, ^ V, a, - P3 a^. 

Like bendino- moments, twistincr moments are nsnally ex- 
pressed in inch -pounds. 

Jl^camjjJe. Let a^ = a., = a^ = 15 inches, a, = 30 inches, 
Pj = 400 pounds, P_, = 500 pounds, P.^ = 750 pounds, and P^ = 
600 pounds. "'=" AVhat is the value of the greatest twisting moment 
in the shaft ? 

At any section between the first and second pulleys, the 
twistino; moment is 

400- X 15 = 6,000 inch-pounds; 

at any section between the second and third it is 

^ 400 X 15 -f 500 X 15 = 13,500 inch-pounds; and 
at any section between the third and fourth it is 

400 X 15 - 500 X 15 - 750 X 30 == - 0,000 inch-pounds. 
Hence the greatest value is 13,500 inch-pounds. 

91. Torsional Stress. The stresses in a twisted shaft are 
called '"torsional'' stresses. The torsional stress on a cross-section 
of a shaft is a shearing stress, as in the case illustrated by Fig. 55, 
which represents a flange coupling in a shaft. AVere it not for 
the bolts, one flange would slip over the other when either part 
of the shaft is turned; but the bolts prevent the slipping. Obvi- 
ously there is a tendency to shear the bolts off unless they are 
screwed up very tight; that is, the material of the bolts is sub- 
jected to shearing stress. 

Just so, at any section of the solid shaft there is a tendency 
for one part to slip past the other, and to prevent the slipping or 

* Xote. Tliese uiimbers were so chosen that the moment of P (driving 
moment) eqnals the sum of the moments of the other forces. This is always 
the ease in a shaft rotating at constant speed; that is, the po^Yer given the 
shaft equals the power taken off. 



STRE^^GTH OF MATERIALS 



109 



sliearin^ of the sliaft, there arise shearing stresses at all parts of 
the cross-section. The shearing stress on the cross-section of a 
shaft is not a uniform stress, its value per unit-area being zero at 
the center of the section, and increasino; toward the circumference. 
In circular sections, solid or hollow, the shearing stress per unit- 
area (unit-stress) varies directly as the distance from the center 
of the section, provided the elastic limit is not exceeded. Thus, 
if the shearing unit-stress at the circumference of a section is 







Fig. 55. 

1,000 pounds j)er square inch, and the diameter of the shaft is 
2 inches, then, at -J inch from the center, the unit-stress is 500 
pounds j)er square inch; and at [ inch from the center it is 2.")0 
j)ounds per square inch. In Fig. 55 tlie arrows indicate the 
values and the directions of the shearing stresses on very <mall 
portions of the cross-section of a shaft there represented. 

92. Resisting Moment. ])y 'M-esisting moment" at a sec- 
tion of a shaft is meant the sum of tlie moments of the shcarinn; 
stresses on tlie ci'oss-section about the axis of tlie sliaft. 

l^'t S^ (leiioti^ th(^ value of the sheaiMiig stress j)er unit-area 
(unit-stress) at the outer ])oints of a section of a. shaft! d the 
dianieter of the section (outsider diameter if the shaft is hollowU 
and (1 ^ th(^ inside diameter. 'J'lieii it can he show n that the re- 
sistin(£ moment is: 

For a solid section, ().I<)();^ S^ (T'\ 

V 1 n - 0.11)(>:^S^ ,./' .//) 

lM)i' a hollow sect ion, 

il 

93. I ornuila for the Strength of a >haft. As in the ease 



110 STEEISTGTII OF MATEKIALS 

of beams, tlie resisting moment equals the twisting moment at 
any section. If T be used to denote twisting moment, then we 
liave the formulas : 

For solid circular shafts, 0.1963 S^cP = T', ] 

lor hollow circular shafts, -^ !— = 1. ( \^o; 

a ) 

In any portion of a shaft of constant diameter, the unit- 
shearing stress Sg is greatest where the twisting moment is greatest. 
Hence, to compute the greatest unit-shearing stress in a shaft, 
w^e first determine the value of the Greatest twisting moment, 
substitute its value in the first or second equation above, as the 
case may be, and solve for Sg. It is customary to express T in 
inch-pounds ar.d the diameter in inches, Sg then being in pounds 
per square inch. 

Exain]}les, 1. Compute the Value of the greatest shearing 
unit- stress in the portion of the shaft betw^een the first and second 
pulleys represented in Fig. e54, assuming values of the forces and 
pulley radii as given in the example of article 90. Suppose also 
that the shaft is solid, its diameter beingr 2 inches. 

The twisting moment T at any section of the portion between 
the first and second pulleys is 0,000 inch-pounds, as shown in the 
example referred to. Hence, substituting in the first of the two 
formulas 15 above, we have 

0.1963 S3 X 2^ = 6,000; 

or, Sg = , ^ ^',.^ -7- = 3,820 pounds per square inch. 

This is the value of the unit-stress at the outside portions of all 
sections between the first and second pulleys. 

2. A hollow shaft is circular in cross-section, and its outer 
and inner diameters are 16 and 8 inches respectively. If the 
working strength of the material in shear is 10,000 pounds per 
square inch, what twisting moment can the shaft safely sustain ? 

The problem requires that w^e merely substitute the values of 
Sg, (1^ and d^ in the second of the above formulas 15, and solve for 
T. Thus, 

.^ ^ O.lOns X 10 000(10' -80 ^ 7,537,920 inch.ponods. 

16 ^ 



STKEN^GTH OF MATEEIALS 111 

EXAMPLES FOR PRACTICE. 

1. ('()iii])iite tlie greatest value of tlie Hliearino- iinit-stivss in 
tlie shaft represented in Fig. 54, using the values of the forces 
and pulley radii given in the example of article 00. the diameter 
of the shaft l)eincT 2 inches. 

Ans. 8,51)5 pounds per square inch 

2. A solid shaft is circular in cross-section and is 9.G inches 
in diameter. If the working strencrth of the material in shear is 
10,()()0 pounds per square iiK'h, how large a twisting moment can 
the shaft safely sustain? (The area of the cross-section is practically 
the same as that of the hollow shaft of example 2 preceding.) 

Ans. 1,736,736 inch-pounds. 

94. Formula for the Power Which a Shaft Can Transmit. 
The p)Ower that a shaft can safely transmit depends on the shear- 
ing working strength of the material of the shaft, on the size of 
the cross- section, and on the speed at which the shaft rotates. 

Let ri denote the amount of horse-power; S^ the shearincr 
working strength in pounds per square inch; d the diameter 
(outside diameter if the shaft is hollow) in inches; d^ the inside 
diameter in inches if the shaft is hollow; and n the number of 
revolutions of the shaft per minute. Then the relation between 
power transmitted, unit-stress, etc., is: 



For solid shafts, II -~ ^ ^^^ , ^^ , 
r or iiollow shatts, li = 



(i6) 



321,()()0./ I 

ExainjilcH. 1. AVhat horse-power can a hollow shaft !(> 
inches and S inches in diameter safely transmit at 50 rcNolutions 
per minute, if the shearincr workinix strencrth of the material is 
10,000 pounds per square inch? 

We have merely to substitute in the second of the two for- 
mulas 16 above, and reduce. Thus, 

10,000 ( 16* -SM 50 
11 = r^21,000~X~lTr^~ ^ *'" horse-pow(M- ^nearlyV 

2. AVhat size of solid shaft is needed to transmit (>,(MM) hor^;e- 
])Ower at 50 revolutions per minuti^ if tlie slu>aring workine; 
Btnuigth of tlie material is 10,000 jjonnds jter si]nare ineh^ 



112 STKENGTH OF MATEKIALS 

"We liave merely to substitute in the iii'st of the two formulas 
10, and solve for d . Thus, 

10,000 X >P X 50 
^'*^^*^ = 321,000 . ' 

,. f 73 6,000 X 321,000 ; 

therefore d = i^^opO X 50 = ^'^^^^ 



or 



d -- 1^8,852 = 15.08 inches. 

(A solid shaft of this diameter contains over 25% more material than 
the hollow shaft of example 1 preceding. There is therefore considerable 
saving of material in the hollow shaft.) 

3. A solid shaft 1 inches in diameter transmits 200 horse- 
power while rotating at 200 revolutions per minute. AYhat is the 
greatest shearing unit-stress in the shaft? 

We have merely to substitute in t]ie first of the equations 10, 
and solve for S^. Thus, 



200 = 



S, X 4^ X 200 
821,000 ' 



200 X 321,000 
or, b = ,3 t^.., = o,01o pounds per square inch. 

EXAMPLES FOR PRACTICE. 

1. What horse-power can a solid shaft 9.0 inches in diameter 
safely transmit at 50 revolutions per minute, if its shearing work- 
ing strength is 10,'000 pounds per square inch ? 

Ans. 1,378 horse-power. 

2. What size of solid shaft is required to transmit 500 horse- 
power at 150 revolutions per minute, the shearing working strength 
of the material being 8,000 pounds per square inch. 

Ans. 5.1 inches. 

3. A hollow^ shaft whose outer diameter is 14 and inner 0.7 
inches transmits 5,000 horse-power at 00 revolutions per minute. 
What is the value of the greatest shearing unit-stress in the shaft? 

Ans. 10,273 pounds per square inch. 

STIFFNESS OF R0D5, BEAMS, AND SHAFTS. 

The preceding discussions have related to the 8trenfjtli of 



STEENGTII OF MATERIALS 113 

materials. We shall now consider principally the cloiKjiit'inu of 
r<)<h^ deflection ()f hea III s^ and tv:i'<t of sJtafts. 

95. Coefficient of Elasticity. According to Ilooke's Law 
(xlrt. 1), p. 7), the elongations of a rod subjected to an increasing 
pull are proportional to the pull, provided that the stresses due to 
the pull do not exceed the elastic limit of the material. AVithin 
the elastic limit, then, the ratio of the pull and the elono-ation is 
constant; hence the ratio of the unit-stress (due to the pullj to the 
nnit-eloncration is also constant. This last-named ratio is called 
'' coefficient of elasticity." If E denotes this coefticient, S the 
unit-stress, and .s- the unit-deformation, then 

^ = V- ("7) 

Coefficients of elasticity are usually expressed in pounds per sciuareinch. 

The preceding remarks, definition, and formula a])plv also to 
a case of compression, provided that the material being compressed 
does not bend, but simply shortens in the direction of the com- 
pressing forces. The following table giv^es the aceraye values of 
the coefficient of elasticity for various materials of construction: 

TABLE Q. 

Coefficients of Elasticity. 



Material. 


Average Coefficient of Elasticity. 


Steel 

Wrou<^ht iron 

Cast iron 

Timber 


30,000,000 pounds per sciuare inch. 
'27,r)00,(X)0 
15,000,000 
1,800,000 " " " 



The eocfRciciits of elasticity for steel ami wrouulit iron, for diiTorent 
j^rades of those materials, are remarkably constant ; but for different grades 
of cast iron the coefficients range from about 10,000,000 to 30,000,000 pimnds 
per square inch. Naturally the coefTicient has not the same value for the 
different kinds of Avood ; for the princii>al woods it ranges from l.()00,000 
(for spruce) to 2,100,000 (for Avliite oak). 

l^'ornuila 17 can Ix^ |)nt in n foi-in more convenient for nse, as 
follows : 

Let 1* denote tiu^ force ])ro(hK'ino- tlit> defiu-mation ; .\ \\w 
area of the cross-section of tlu^ ])iece on which V nets ; / the length 
of the pii'CA^ ; and 1) the dtd'ormatiou (^elongation or shoiteningV 



lU STREKGTH OF MATERIALS 

Then 

S = P -^ A (see eqtiation 1), 
and s = D -^ I (see equation 2). 

Hence, substituting these values in equation 17, we have 

E = ^;orD=^- (17) 

The first of these two equations is used for computing the value of 
tlie coefficient of elasticity from measurements of a '' test," and 
the second for computing the elongation or shortening of a given 
rod or bar for which the coefficient is known. 

Exain]L)le^. 1. It is required to compute the coefficient of 
elasticity of the material the record of a test of which is given on 
page 9. 

Since the unit-stress S and unit-elongation s are already 
computed in that table, we can use equation IT instead of the first 
of equations IT'. The elastic limit being between 40,000 and 
45,000 2:)0unds per square inch, we may use any value of the 
unit-stress less than that, and the corresponding unit-elongation„ 

Thns, with the first values mven, 

5.000. 
E = OlKMT = 2'.K400,000. 



With the second. 



10,000 



This lack of constancy in the value of E as computed from different 
loads in a test of a given material, is in part due to errors in measuring the 
deformation, a measurement difficult to make. The value of the coefficient 
adopted from such a test, is the average of all the values of E which can be 
computed from the record. 

2. How much will a pull of 5,000 pounds stretch a round 
steel rod 10 feet lona and 1 inch in diameter ? 

We use the second of the two formulas IT'. Since A = 
0.7854 X 1' = 0.T854 square inches, I = 120 inches, and E = 
30,000,000 pounds per square inch, the stretch is: 

^ 5,000 X 120 „ „„^. . , 

^ = ' 0.7854 X 30,000,000 = ^'^^^^ ^"^^^ 



STRENGTH OF MATEEIALS Hi 



EXAMPLES FOR PRACTICE. 

1. What is the coefficient of elasticity of a material if a pull 
of 20,000 nounds will stretch a rod 1 inch in diameter and 4 feet 



V 



p 



loner 0.045 inch ? 

Ans. 27,000,000 pounds per square inch. 

2. IIow much will a pull of 15,000 pounds elongate a round 
cast-iron rod 10 feet lonp; and 1 inch in diameter ? 

Ans. 0.152 inch. 

96. Temperature Stresses. In the case of most materials, 

when a bar or rod is heated, it lencrthens; and when cooled, it 

shortens if it is free to do so. The coefficient of linear expansion 

of a material is the ratio which the elongation caused in a rod or 

bar of the material by a change of one degree in temperature bears 

to the length of the rod or bar. Its values for Fahrenheit degrees 

are about as follows : 

For Steel, 0.0000065. 

For Wrought iron, .0000067. 
For Cast iron, .0000062. 

Let K be used to denote this coefficient; t a change of tern- 
perature, in degrees Fahrenheit; I the length of a rod or bar; 
and J) the change in length due to the change of temperature. 
Then 

D = X / /. (18) 

D and I are expressed in the same unit. 

If a rod or bar is confined or restrained so that it cannot 
change its length wlien it is^ heated or cooled, then any clianov in 
its temperature produces a stress in the rod; such are called tem= 
perature stresses. 

J'Jxamjdes. 1. A steel rod connects two solid A\alls and is 
screwed up so that the unit-stress in it is 10.000 ])()unds i)er 
square inch. Its temperature falls 10 degrees, and it is obserwd 
that the walls have not been drawn together. AVliat is the tein|HM-- 
ature stress produced by the change of temperatur(\ and \\liat is 
the actual unit-stress in the rod at tlu' new teiujuM'atnre i 

Let / denote the lengtli of tlie rod, TIumi the ehanoc in 
length which would occur if the I'od wei'i^ free, isgi\en 1)\ forniuhi 
IS, al)()\'e, tlms: 

D ^ 0()0()()()(;5 -, 10 X / =~ o.()0()()(;5 /. ' 



116 STEENGTH OF MATEKIALS 



Now, since tlie rod could not shorten, it lias a greater than normal 
length at the new temperature; that is, the fall in temperature has 
produced an effect equivalent to an elongation in the rod amount- 
ing to D, and hence a tensile stress. This tensile stress can be 
computed from the elongation D by means of formula 17. Thus, 

S = E s; 

and since .<?, the unit-elongation, equals 

D .0000985 J 



I " I 



= .0000035, 



S = 30,000,000 X .0000065 == 195.0 pounds per square inch. 
This is the value of the temperature stress; and the new unit- 
stress equals 

10,000 -f- 195.0 = 10,195 pounds per square inch. 

Notice that the unit temperature stresses are independent of the length 
of the rod and the area of its cross-section. 

2. Suppose that the change of temperature in the preceding 
example is a rise instead of a fall. What are the values of the 
temperature stress due to the change, and of the new unit-stress in 
the rod ? 

The temperature stress is the same as in example 1, that is, 
1,950 pounds per square inch ; but the rise in temperature 
releases, as it were, the stress in the rod due to its being screwed 
up, and the final unit stress is 

10,000 - 1,950 = 8,050 pounds per square inch. 

EXAHPLE FOR PRACTICE. 

1. The ends of a wrought-iron rod 1 inch in diameter are 
fastened to two heavy bodies which are to be drawn together, the 
temperature of the rod being 200 degrees when fastened to the ob- 
jects. A fall of 120 degrees is observed not to move them. 
What is the temperature stress, and what is the pull exerted by 
the rod on each object ? 

\ Temperature stress, 22,000 pounds per square inch. 
Ans. I p^^^^^ ^^^280 pounds. 

97. Deflection of Beams. Sometimes it is desirable to know 
how much a given beam will deflect under a given load, or to design 



STlU^NGTll OF :MATEiaALS 117 

a beam which will not deflect more than a certain amount under a 
given load. In Table B, page 55, Pait I, are given formulas for 
deflection in certain cases of beams and different kinds of loading. 

In those formulas, fi denotes deflection; I the moment of inertia of tlie 
cross-section of the beam with respect to the neutral axis, as in equation 6 ; 
and E the coefficient of elasticity of the material of the beam (for values, see 
Art. 95). 

In each ease, the load should be expressed in pounds, the length in 
inches, and the moment of inertia in biquadratic inches; then the deflection 
will be in inches. 

According to the formulas for d ^ the deflection of a beam 
varies inversely as the coefficient of its material (E) and the mo- 
ment of' inertia of its cross-section (T) ; also, in the first four and 
last two cases of the table, the deflection varies directly as the cube 
of the length (P). 

Examjple. What deflection is caused by a uniform load of 
0,400 pounds (including weight of the beam) in a wooden beam 
on end supports, which is 12 feet long and 6 X 12 inches in 
cross -section ? (This is the safe load for the beam ; see example 
1, Art. 05.) 

The formula for this case (see Table B, page 55) is 

^_ 5 WP 

'•^ ~ 884 EI • 

Here W -= 0,400 pounds ; I ^ \\\ inches ; E --. 1,800,000 
pounds per square inch ; and 

1 , 1 

I = T9 ^>'^ = 19 <> X 12^== 804 inchest 

Ilencc^ th(^ (h'flection is 

— ___ ^ X 0,400 X 144^ __ 

'^ ~ ^sr'xXsooXm'Tx^soI "" ^^•^'* '"^*^'- 

EXAMPLES FOR PRACTICE. 

1. (compute tlie deflection of a timber l)uilt-in cantilever 
8x8 inches which projects 8 feet fi-oiii the wall and l)ears an 
end h)ad of UOO ])()unds. (Tliis is the safe lond for tlie t'antilever. 
see example 1, Art. 05.) 

A us, 0,48 inch. 

2. Compute thy defieetit)ii caused l)\ a nnifonu load of 10,000 



118 



STREXGTII OF MATERIALS 



pounds on a 42-pound 15-inch steel I-beam whicli is 16 feet long 
and rests on end sn])ports. 

Ans. 0.28 inch. 

98. Twist of Shafts. Let Fig, 57 represent a portion of a 

shaft, and suppose that the part represented lies wholly between 




Fig. 57. 

two adjacent pulleys on a shaft to which twisting forces are applied 
(see Fig. 51). Imagine two radii '//ta and /Jj in the ends of the 
portion, they being parallel as shown when the shaft is not twisted. 
After the shaft is twisted they will not be parallel, ma having 
moved to ma\ and /t7j to nh'. The ano-le between the two linec in 
their twisted positions (^/na' and n7/^ is called the angle of twist, 
or angle of torsion, for the length /. If a'a" is parallel to al, then 
the angle a"rib' equals the angle of torsion. 

If the stresses in the portion of the shaft considered do not 
exceed the elastic limit, and if the twistino* moment is the same 
for all sections of the portion, then the angle of torsion a (in 
degrees) can be computed from the following: 

For solid circular shafts. 



a = 



584 T/ 36.800.000 H/ 



E^ (I' ~ 
For hollow circular shafts, 
584 TId 



a == 



E' (I'll 



36,800,000 HZ 



Y 



(19) 



J 



E\d'-d^y Y} {d' ~d,') n 

Here T, Z, rZ, ^-Zi, H, and n have the same meanings as in Arts. 93 
and 94, and should be expressed in the units there used. The 
letter E^ stands for a quantity called coefficient of elasticity for 
shear; it is analogous to the coefiicient of elasticity for tension and 
compression (E), Art. 95. The values of E^ for a few materials 
average about as follows (roughly E^ = | E) : 



STEENGTII OF MATERIALS 119 

Pot 'Steel, 11,000,(XX) pounds per s(iuare inch. 

For Wrought iron, 10,000,000 " " " " 

For Cast iron, 6,000,000 " " » " 

Exaw.ple. What is the value of the angle of torsion of a 
steel shaft 60 feet long when transmitting 6,000 horse-power at 
50 revolutions per minute, if the shaft is hollow and its outer and 
inner diameters are 16 and 8 inches respectively ? 

Here I = 720 inches; hence, substituting in the appropriate 
formula (10), we find that 

36,800,000 X 6,000 x '^SO , ^ , 
^ = 11,000,000 X (16- - 8-) 50 = ^- ' ^''^'''^'' 

EXAMPLE FOR PRACTICE. 

Suppose that the first two pulleys in Fig. 51 are 12 feet 
apart; that the diameter of the shaft is 2 inches; and that Pj = 400 
pounds,-and a^ = 15 inches. If the shaft is of wrought iron, 
what is the value of the angle of torsion for the portion between 
the first two pulleys? 

Ans. 3.15 degrees. 

99. Non=eIastic Deformation. The preceding formulas for 
elongation, deflection, and twist hold only so long as the greatest 
unit-stress does not exceed the elastic limit. There is no theory, 
and no formula, for non -elastic deformations, those corresponding 
to stresses which exceed the elastic limit. It is well known, how- 
ever, that non -elastic deformations are not proportional to the 
forces producing them, but increase much faster than the loads. 
The value of the ultimate elongation of a rod or bar (that is, the 
amount of elongation at rupture), is quite well known for many 
materials. This elongation, for eight-inch specimens of various 
materials (see Art. 16), is : 

For Cast iron, about 1 per cent. 
For Wrought iron (phites), 12 - 15 per cent. 
For " " ( bars), 20 - 25 " " . 

For Structural steel, 22 - 2(j " " . 

Specimens of ductile materials (such as wrought iron and 
fitrnctural steel), wlicn pulled to destruction, neck down, that is, 
diiiiinisli very considcirably in cross-section at soim^ place along 
the length of the specimen. The decrease in cross-sectional area 



120 



BTKENGTH OF MATEKIxlLS 



is known as reduction of area, and its value forwroucrht iron and 
steel may be as niiicli as 50 per cent. 

RIVETED JOINTS. 

loo. Kinds of Joints. A lap joint is one in which the 
plates or bars joined overlap each other, as in Fig. 58, a. A butt 
joint is one in which the plates or bars that are joined butt against 
each other, as in Fig. 58, h. The thin side plates on butt joints 



p-^ 






<^^, 



P- — \ , ' ' h ' ' t — ^P 



I , I'l 






rr 



— >p 



p* 




Fig. 58. 

are called cover=plates ; the thickness of each is always made not 
less than one-half the thickness of the main plates, that is, the 
plates or bars that are joined. Sometimes butt joints are made 
with only one cover-plate; in such a case the thickness of the 
cover-plate is made not less than that of the main plate. 

"When wide bars or plates are riveted together, the rivets are 
placed in rows, always jiarallel to the •• seam " and sometimes also 
perpendicular to the seam; but when we speak of a row of rivets, 
we mean a row parallel to the seam. A lap joint Avith a single 
row of rivets is said to be single=riveted ; and one with two rows 
of rivets is said to be double=riveted. A butt joint with two rows 
of rivets (one on each side of the joint) is called '' single-riveted," 
and one with four rows (two on each side) is said to be '• double- 
riveted." 

The distance between the centers of consecutive holes in a 
row of rivets is called pitch. 

loi. Shearing Strength, or Sher^ring Value, of a Rivet. 
"When a lap joint ia subjected to tension (that is, when P, Fig. 58, 
a^ is a pull ), and when the joint is subjected to compression (when 
P is a push), there is a tendency to cut or shear each rivet along 
the surface between the two plates. In butt joints with two cover- 



STRENGTH OF MATERIALS 121 

plates, there is a tendency to cut or shear each rivet on two sur- 
faces (see Fig. 58, ^). Therefore the rivets in the lap joint are 
said to be in single shear ; and those in the butt joint (two covers) 
are said to be in double shear. 

The " shearincr value " of a rivet means the resistance which 
it can safely offer to forces tendino- to shear it on its cross-section. 
This value depends on the area of the cross-section and on the work- 
inp; streno'th of the material. Let J denote the diameter of the 
cross-section, and S^ the shearing working strength. Then, since 
the area of the cross-section equals 0.7854 J\ the shearing strength 
of one rivet is : 

For single shear, 0.7854: d- Ss . 

For double shear, 1.5708 c/2 Ss . 

102. Bearing Strength, or Bearing Value, of a Plate. AVhen 
a joint is subjected to tension or compression, each rivet ])resse3 
against a part of the sides of the holes through whicli it passes. 
By "bearing value" of a plate (in this co>]nection) is meant the 
pressure, exerted by a rivet against the side of a hole in the plate, 
which the plate can safely stand. I'll is value depends on the 
thickness of the ])late, on the diameter of the rivet, and on the 
comj)ressive working strength of the j)late. Exactly how it 
de])ends on these three (pialities is not known; 1)ut the beai'ing 
value is alwa\s computed from the ex[)ression / // S^., wherein / 
denotes the thickness of the ])late; /'/, the diameter of the rivt't or 
hole; and S^, the workino; stren(£th of the plate. 

103. Frictional Strength of a Joint. AVhen a joint is sub 
jected to tension or compression, there is a tendency to slippage 
between the faces of the plates of tho joint. This tendency is 
overcome wholly oi- in part by frictional resistance between the 
plates. The frictional resistance in a well-made joint may be 
very large, for i-ivets art^ j)ut into a joint hot, and are headed or 
capped before being cooled. In cooling they contract, di'awing the 
j)lates of the joint tightly against each other, and producing a 
great pressure between them, which gives the joint a coi-i'espond- 
ingly large frictional strength. It is the opinion (►f some tliai 
all well-mad(^ joints perform their service by nu'ans of tluir 
frictional strength; that is to say, tlu^ rivets act onl\ 1»\ pi-cssing 
ihci plates together and ai'i^ not undi'r slu'aring stress, iu)r 



122 STREIS^GTH OF MATERIALS 

are the plates under compression at the sides of their holes. The 
" frictional strength" of a joint, however, is usually regarded as 
uncertain, and generally no allowance is rnade for friction in com- 
putations on the strength of riveted joints. 

104. Tensile and Compressive Strength of Riveted Plates. 
The holes punched or drilled in a plate or bar weaken its tensile 
strength, and to compute that strength it is necessary to allow for 
the holes. By net section, in this connection, is meant the small- 
est cross-section of the plate or bar ; this is always a section along 
a line of rivet holes. 

If, as in the foreo^oingr article, t denotes the thickness of the 
plates joined,; cZ, the diameter of the holes; n^^ the number of riv- 
ets in a row ; and ?/', the width of the plate or bar; then the net 
section = {w — n^T) t. 

Let Sj denote the tensile working strength of the plate ; .then 
the strength of the unriveted plate is wt"^^^ and the reduced tensile 
strength is [^10 — n/I') t S^. 

The compressive strength of a plate is also lessened by the 
presence of holes ; but when they are again filled up, as in a joint, 
the metal is replaced, as it were, and the compressive strength of 
the plate is restored. Xo allowance is therefore made for holes in 
figuring the compressive strength of a plate. 

105. Computation of the Strength of a Joint. The strength 
of a joint is determined by either (1) the shearing value of the 
rivets; (2) the bearing value of the plate; or (3) the tensile 
strength of the riveted plate if the joint is in tension. Let P^ de- 
note the. strength of the joint as computed from the shearing 
values of the rivets ; P^, that computed from the bearing value of 
the plates ; and P^, the tensile strength of the riveted plates. 
Then, as before explained, 

Pt= [vj - n^d) t^^\ \ 

P3=77, 0.7854^^8,; and [ (20) 

n^ denoting the total number of rivets in the joint ; and n^ denot- 
ing the total number of rivets in a lap joint, and one-half the 
number of rivets in a butt joint. 

Examples. 1. Two half -inch plates 11 inches wide are con- 



STEENGTII OF MATERIALS 123 

nected by a single lap joint double-riveted, six rivets in two rows. 
If the diameter of the rivets is | inch, and the working strengths 
are as follows : 8^-= 12,000, 8^= 7,500, and S^= 15,000 pounds 
per square inch, what is the safe tension which the joint can 
• transmit ? 

Here n^^=^ 3, n=^ 6, and n^= 6 ; hence 

Pt= (^-y ~ '^ X X^ ^ T ^ -^^'^^^ ^ ^^'^^^ pounds; 
P,-: 6 X 0.785J: X {-^y X 7,500 = 19,880 pounds ; 

1 3 

' p^=: X ^ X -J- X 15,000 = 33,750 pounds. 

Since P^ is the least of these three values, the strength of the 
joint depends on the shearing value of its rivets, and it equals 
19,880 pounds. 

2. Suppose that the plates described in the preceding example 
are joined by means of a butt joint (two cover-plates), and 12 
rivets are used, being spaced as before. What is the safe tension 
which the joint can bear ? 

Here oi^ = 3, n^ = 12, and ^3 = 6; hence, as in the preced- 
ing example, 

1\ = 31,500; and P, -- 33,750 pounds; but 

P, = 12 X 0.7854 X {^y X 7,500 =■. 39,760 pounds. 

The strength equals 31,500 pounds, and the joint is stronger than 
the first. 

3. Suppose that in the preceding example the rivets are 
arranged in rows of two. What is the tensile strength of the 
joint ? 

Here ?Aj = 2, ?/,^ = 12, and n,, = 6; hence, as in the preced- 
ing example, 

P3 == 39,7()0; and ]\ =^ 33,750 pounds; but 

P^ = (7 -^.— 2 X -',-) -4- X 12,000 ^- 'MS)0() pounds. 

The strength equals 33,750 pouiuls, and this joint is stronger than 
eitlier of the first two. 



124 STREXGTH OF MATERIALS 



EXAMPLES FOR PRACTICE. 

Note. Use working- strengths as in example 1, above. 
St = 12,000, Ss = 7,500, and Sc = 15,000 pounds per square inch. 

1. Two half-iiich plates 5 inclies wide are connected l)y a 
lap joint, with two |-inch rivets in a row. What is the safe 
strength of the joint ? 

Ans. 6,625 pounds. 

2. Solve the preceding example supposing that four |-inch 
rivets are used, in two rows. 

Ans. 13,250 pounds. 

3. Solve example 1 supposing that three 1-inch rivets are 

used, placed in a row lengthwise of the joint. 

Ans. 1T,6T0 pounds. 

4. Two half-inch plates 5 inches wide are connected by a 
butt joint (two cover-plates), and four J-inch rivets are used, in 
two rows. What is the strength of the loint ? 

Ans. 11,250 pounds. 

io6. Efficiency of a Joint. The ratio of the strength of a 
joint to that of the solid plate is called the " efficiency of the 
joint." If ultimate strengths are used in computing the ratio, 
then the efficiency is called ultimate efficiency; and if working 
strengths are used, then it is called working efficiency. In the 
followin cr we refer to the latter. An efficiency is sometimes ex- 
pressed as a jjer cent. To express it thus, multiply the ratio 
strengtJt of joint -^ strength of solid 2)^ ('■t<-'^ by l^^- 

Exaiivple. It is required to compute the efficiencies of the 
joints described in the examples worked out in the preceding article. 

In each case the plate is \ inch thick and 7^ inches wide; 
hence the tensile working strength of the solid plate is 

7-i- X 4- X 1-^000 = 45,000 pounds. 



TherefvOre the efficiencies of the joints are : 

19,880 ^,, \, 

— ^ = 0.44, or 44 per cent; 

45,000 ' ^ 

31,500 ^.^ .. 

— 1— — = 0. /O, or tij per cent; 
45,000 ' ^ 

33,750 ... ^. 



(1) 
(2) 



INDEX 



Page 

Angle of torsion 118 

Angle of twist 118 

Applications of first beam formula ■. 01 

Beam formula 

first 61 

second 75 

Beams 21 

deflection of 116 

stiffness of 112 

Bearing strength of a plate 121 

Bending moment 25, 34 

maximum 42 

Brick, ultimate strength of 17 

Broken straight -line formula . .■ . 103 

Built-up sections, moment of inertia of 51 

Butt joint 120 

Cast iron 

intension 15 

in compression 17 

Center of gravity of an area 43 

Center of gravity of built-up sections 46 

Centrally loaded column 90 

Coefficient of elasticity 113 

for shear 118 

Coefficient of linear expansion 115 

Column formulas 

graphical represent al ion of i)5 

Rankine's ' \)() 

Column loads, kinds of 90 

Columns 

classes of SS 

cross-sections of 88 

design of 1 04 

end conditions of 87 

strength of 87 

Combination column formulas 90 

Combined flexural and direct stress 85 

Compression, materials in 15 

C'oiiipr('ssi\'c si HMigt li of ii\('t cd plnl (\^ 122 

T'onl ill nous b(>ain • 21 



126 INDEX 

Page 

Cover-plates 1 20 

Deflection of beams ." 116 

Deformation q 

Design of 

columns 104 

timber beams 78 

Direct stress 81 

Double shear 121 

Eccentrically loaded 90 

Efficiency of a joint , 124 

Elastic limit 7 

Elasticity , 6 

End conditions of column 87 

Euler formulas 96 

External shear 25 

Factor of safety 10 

Fibre stresses 57, 59 

First beam formula 61 

Flexural stress 81 

Flexure and compression 83 

Flexure and tension 81 

Formula for power which shaft can transmit Ill 

Formula for strength of shaft 109 

Frictional strength of joint 121 

Graphical representation of column formulas 95 

Hooke's law 7 

Horizontal shear 77 

Inclined forces 56 

Joint 

efficiency of 124 

frictional strength of 121 

Kinds of loads and beams 54, 81 

Kinds of stress 4, 57 

Lap joint 120 

Laws of strength of beams 73 

Linear expansion, coefficient of 115 

Longitudinal forces 56 

Main plates 120 

Materials 

in compression 15 

in shear 17 

in tension " 13 

Maximum bending moment 42 

Maximum shear 33 

Metals in shear 18 

Modulus of rupture . 74 

Moment diagrams 38 

Moment of a force 18 



INDEX 127 

Page 

Moment of inertia 48 

of built-up sections 51 

of a rectangle 50 

unit of 49 

Moments, principle of 19 

Neutral axis ". 56 

Neutral line 56 

Neutral surface 56 

Non-elastic deformation 119 

Notation 26, 35 

Parabola-Euler formulas 100 

Posts 87 

Principle of moments 19 

applied to areas ' *. 44 

Radius of gyration 88 

Rankine's column formula 90 

Reactions of supports 18 

Rectangle, moment of inertia of 50 

Reduction formula 50 

Resisting.moment 59, 109 

value of 60 

Resisting shear 75 

Restrained beam 21 

Rivet, shearing strength of 120 

Riveted j oints 1 20 

Rods, stiffness of 112 

Rule of signs 25, 34 

Safe load of a beam 73 

Safe strength of a beam 73 

Second beam formula 75 

Section modulus v 61 

Shafts 

stififnese of 112 

strength of 107 

twist of 118 

twisting moment of . 107 

Shear diagrams . 29 

Shear stress 4 

Shearing strength of rivet 120 

Shears, units for 26 

Signs, rule of 25, 3 1 

Simple beam 21 

Simple stress , 3 

Single shear 121 

Steel in compression 16 

Steel in tension 11 

Stiffness of rods, beams, and shafts 112 

Stone, ull inialc strength oC 17 



128 INDEX 

Page 

Straight-line formulas ~ 96 

Strength of beams 54 

laws of 73 

Strength of columns 87 

Strength of joint, computation of . 122 

Strength of shafts 107 

Stress, kinds of 4 

Stress-deformation diagram 8 

Struts 87 

Tables 

bending moment 55 

coefficients of elasticity 113 

deflection 55 

factors of safetj' 12 

maximum shear, values of . ■ -. *...... 55 

mild steel columns, data for 97, 101 

moduli of rupture 74 

moment diagrams 55 

moments of inertia 54 

properties of standard I-beams a . . 72 

radii of gyration 54 

section moduli 54 

shear diagrams 55 

Temperature stresses 115 

Tensile strength of riveted plates 1 22 

Tension, materials in -. . . . 13 

Timber 

in compression IG 

in shear 17 

in tension. ; 13 

Timber beams, design of 78 

Torsional stress 1 OS 

Transverse forces 54 

Twist of shafts 118 

Twisting moment of shaft 107 

Ultimate strength 8 

Unit-deformation 6 

Unit of moment of inertia ? 49 

Unit-stress 5 

Units 35 

Units for shears. 26 

Value of resisting moment 60 

AVrought-iron 

in compression 16 

in tension 14 

Working strength 10 

Working stress 10 



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